Miscellaneous arithmetic functions#
AUTHORS:
Kevin Stueve (2010-01-17): in
is_prime(n)
, delegated calculation ton.is_prime()
- sage.arith.misc.CRT(a, b, m=None, n=None)#
Return a solution to a Chinese Remainder Theorem problem.
INPUT:
a
,b
- two residues (elements of some ring for which extended gcd is available), or two lists, one of residues and one of moduli.m
,n
- (default:None
) two moduli, orNone
.
OUTPUT:
If
m
,n
are notNone
, returns a solution \(x\) to the simultaneous congruences \(x\equiv a \bmod m\) and \(x\equiv b \bmod n\), if one exists. By the Chinese Remainder Theorem, a solution to the simultaneous congruences exists if and only if \(a\equiv b\pmod{\gcd(m,n)}\). The solution \(x\) is only well-defined modulo \(\text{lcm}(m,n)\).If
a
andb
are lists, returns a simultaneous solution to the congruences \(x\equiv a_i\pmod{b_i}\), if one exists.See also
EXAMPLES:
Using
crt
by giving it pairs of residues and moduli:sage: crt(2, 1, 3, 5) 11 sage: crt(13, 20, 100, 301) 28013 sage: crt([2, 1], [3, 5]) 11 sage: crt([13, 20], [100, 301]) 28013
You can also use upper case:
sage: c = CRT(2,3, 3, 5); c 8 sage: c % 3 == 2 True sage: c % 5 == 3 True
Note that this also works for polynomial rings:
sage: K.<a> = NumberField(x^3 - 7) sage: R.<y> = K[] sage: f = y^2 + 3 sage: g = y^3 - 5 sage: CRT(1,3,f,g) -3/26*y^4 + 5/26*y^3 + 15/26*y + 53/26 sage: CRT(1,a,f,g) (-3/52*a + 3/52)*y^4 + (5/52*a - 5/52)*y^3 + (15/52*a - 15/52)*y + 27/52*a + 25/52
You can also do this for any number of moduli:
sage: K.<a> = NumberField(x^3 - 7) sage: R.<x> = K[] sage: CRT([], []) 0 sage: CRT([a], [x]) a sage: f = x^2 + 3 sage: g = x^3 - 5 sage: h = x^5 + x^2 - 9 sage: k = CRT([1, a, 3], [f, g, h]); k (127/26988*a - 5807/386828)*x^9 + (45/8996*a - 33677/1160484)*x^8 + (2/173*a - 6/173)*x^7 + (133/6747*a - 5373/96707)*x^6 + (-6/2249*a + 18584/290121)*x^5 + (-277/8996*a + 38847/386828)*x^4 + (-135/4498*a + 42673/193414)*x^3 + (-1005/8996*a + 470245/1160484)*x^2 + (-1215/8996*a + 141165/386828)*x + 621/8996*a + 836445/386828 sage: k.mod(f) 1 sage: k.mod(g) a sage: k.mod(h) 3
If the moduli are not coprime, a solution may not exist:
sage: crt(4,8,8,12) 20 sage: crt(4,6,8,12) Traceback (most recent call last): ... ValueError: no solution to crt problem since gcd(8,12) does not divide 4-6 sage: x = polygen(QQ) sage: crt(2,3,x-1,x+1) -1/2*x + 5/2 sage: crt(2,x,x^2-1,x^2+1) -1/2*x^3 + x^2 + 1/2*x + 1 sage: crt(2,x,x^2-1,x^3-1) Traceback (most recent call last): ... ValueError: no solution to crt problem since gcd(x^2 - 1,x^3 - 1) does not divide 2-x sage: crt(int(2), int(3), int(7), int(11)) 58
crt also work with numpy and gmpy2 numbers:
sage: import numpy sage: crt(numpy.int8(2), numpy.int8(3), numpy.int8(7), numpy.int8(11)) 58 sage: from gmpy2 import mpz sage: crt(mpz(2), mpz(3), mpz(7), mpz(11)) 58 sage: crt(mpz(2), 3, mpz(7), numpy.int8(11)) 58
- sage.arith.misc.CRT_basis(moduli)#
Return a CRT basis for the given moduli.
INPUT:
moduli
- list of pairwise coprime moduli \(m\) which admit anextended Euclidean algorithm
OUTPUT:
a list of elements \(a_i\) of the same length as \(m\) such that \(a_i\) is congruent to 1 modulo \(m_i\) and to 0 modulo \(m_j\) for \(j\not=i\).
Note
The pairwise coprimality of the input is not checked.
EXAMPLES:
sage: a1 = ZZ(mod(42,5)) sage: a2 = ZZ(mod(42,13)) sage: c1,c2 = CRT_basis([5,13]) sage: mod(a1*c1+a2*c2,5*13) 42
A polynomial example:
sage: x=polygen(QQ) sage: mods = [x,x^2+1,2*x-3] sage: b = CRT_basis(mods) sage: b [-2/3*x^3 + x^2 - 2/3*x + 1, 6/13*x^3 - x^2 + 6/13*x, 8/39*x^3 + 8/39*x] sage: [[bi % mj for mj in mods] for bi in b] [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
- sage.arith.misc.CRT_list(values, moduli)#
Given a list
values
of elements and a list of correspondingmoduli
, find a single element that reduces to each element ofvalues
modulo the corresponding moduli.See also
EXAMPLES:
sage: CRT_list([2,3,2], [3,5,7]) 23 sage: x = polygen(QQ) sage: c = CRT_list([3], [x]); c 3 sage: c.parent() Univariate Polynomial Ring in x over Rational Field
It also works if the moduli are not coprime:
sage: CRT_list([32,2,2],[60,90,150]) 452
But with non coprime moduli there is not always a solution:
sage: CRT_list([32,2,1],[60,90,150]) Traceback (most recent call last): ... ValueError: no solution to crt problem since gcd(180,150) does not divide 92-1
The arguments must be lists:
sage: CRT_list([1,2,3],"not a list") Traceback (most recent call last): ... ValueError: arguments to CRT_list should be lists sage: CRT_list("not a list",[2,3]) Traceback (most recent call last): ... ValueError: arguments to CRT_list should be lists
The list of moduli must have the same length as the list of elements:
sage: CRT_list([1,2,3],[2,3,5]) 23 sage: CRT_list([1,2,3],[2,3]) Traceback (most recent call last): ... ValueError: arguments to CRT_list should be lists of the same length sage: CRT_list([1,2,3],[2,3,5,7]) Traceback (most recent call last): ... ValueError: arguments to CRT_list should be lists of the same length
- sage.arith.misc.CRT_vectors(X, moduli)#
Vector form of the Chinese Remainder Theorem: given a list of integer vectors \(v_i\) and a list of coprime moduli \(m_i\), find a vector \(w\) such that \(w = v_i \pmod m_i\) for all \(i\). This is more efficient than applying
CRT()
to each entry.INPUT:
X
- list or tuple, consisting of lists/tuples/vectors/etc of integers of the same lengthmoduli
- list of len(X) moduli
OUTPUT:
list
- application of CRT componentwise.
EXAMPLES:
sage: CRT_vectors([[3,5,7],[3,5,11]], [2,3]) [3, 5, 5] sage: CRT_vectors([vector(ZZ, [2,3,1]), Sequence([1,7,8],ZZ)], [8,9]) [10, 43, 17]
- class sage.arith.misc.Euler_Phi#
Bases:
object
Return the value of the Euler phi function on the integer n. We defined this to be the number of positive integers <= n that are relatively prime to n. Thus if n<=0 then
euler_phi(n)
is defined and equals 0.INPUT:
n
- an integer
EXAMPLES:
sage: euler_phi(1) 1 sage: euler_phi(2) 1 sage: euler_phi(3) 2 sage: euler_phi(12) 4 sage: euler_phi(37) 36
Notice that euler_phi is defined to be 0 on negative numbers and 0.
sage: euler_phi(-1) 0 sage: euler_phi(0) 0 sage: type(euler_phi(0)) <class 'sage.rings.integer.Integer'>
We verify directly that the phi function is correct for 21.
sage: euler_phi(21) 12 sage: [i for i in range(21) if gcd(21,i) == 1] [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]
The length of the list of integers ‘i’ in range(n) such that the gcd(i,n) == 1 equals euler_phi(n).
sage: len([i for i in range(21) if gcd(21,i) == 1]) == euler_phi(21) True
The phi function also has a special plotting method.
sage: P = plot(euler_phi, -3, 71)
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: euler_phi(int8(37)) 36 sage: from gmpy2 import mpz sage: euler_phi(mpz(37)) 36
AUTHORS:
William Stein
Alex Clemesha (2006-01-10): some examples
- plot(xmin=1, xmax=50, pointsize=30, rgbcolor=(0, 0, 1), join=True, **kwds)#
Plot the Euler phi function.
INPUT:
xmin
- default: 1xmax
- default: 50pointsize
- default: 30rgbcolor
- default: (0,0,1)join
- default: True; whether to join the points.**kwds
- passed on
EXAMPLES:
sage: from sage.arith.misc import Euler_Phi sage: p = Euler_Phi().plot() sage: p.ymax() 46.0
- sage.arith.misc.GCD(a, b=None, **kwargs)#
Return the greatest common divisor of
a
andb
.If
a
is a list andb
is omitted, return instead the greatest common divisor of all elements ofa
.INPUT:
a,b
– two elements of a ring with gcd ora
– a list or tuple of elements of a ring with gcd
Additional keyword arguments are passed to the respectively called methods.
OUTPUT:
The given elements are first coerced into a common parent. Then, their greatest common divisor in that common parent is returned.
EXAMPLES:
sage: GCD(97,100) 1 sage: GCD(97*10^15, 19^20*97^2) 97 sage: GCD(2/3, 4/5) 2/15 sage: GCD([2,4,6,8]) 2 sage: GCD(srange(0,10000,10)) # fast !! 10
Note that to take the gcd of \(n\) elements for \(n \not= 2\) you must put the elements into a list by enclosing them in
[..]
. Before trac ticket #4988 the following wrongly returned 3 since the third parameter was just ignored:sage: gcd(3, 6, 2) Traceback (most recent call last): ... TypeError: ...gcd() takes ... sage: gcd([3, 6, 2]) 1
Similarly, giving just one element (which is not a list) gives an error:
sage: gcd(3) Traceback (most recent call last): ... TypeError: 'sage.rings.integer.Integer' object is not iterable
By convention, the gcd of the empty list is (the integer) 0:
sage: gcd([]) 0 sage: type(gcd([])) <class 'sage.rings.integer.Integer'>
- class sage.arith.misc.Moebius#
Bases:
object
Return the value of the Möbius function of abs(n), where n is an integer.
DEFINITION: \(\mu(n)\) is 0 if \(n\) is not square free, and otherwise equals \((-1)^r\), where \(n\) has \(r\) distinct prime factors.
For simplicity, if \(n=0\) we define \(\mu(n) = 0\).
IMPLEMENTATION: Factors or - for integers - uses the PARI C library.
INPUT:
n
- anything that can be factored.
OUTPUT: 0, 1, or -1
EXAMPLES:
sage: moebius(-5) -1 sage: moebius(9) 0 sage: moebius(12) 0 sage: moebius(-35) 1 sage: moebius(-1) 1 sage: moebius(7) -1
sage: moebius(0) # potentially nonstandard! 0
The moebius function even makes sense for non-integer inputs.
sage: x = GF(7)['x'].0 sage: moebius(x+2) -1
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: moebius(int8(-5)) -1 sage: from gmpy2 import mpz sage: moebius(mpz(-5)) -1
- plot(xmin=0, xmax=50, pointsize=30, rgbcolor=(0, 0, 1), join=True, **kwds)#
Plot the Möbius function.
INPUT:
xmin
- default: 0xmax
- default: 50pointsize
- default: 30rgbcolor
- default: (0,0,1)join
- default: True; whether to join the points (very helpful in seeing their order).**kwds
- passed on
EXAMPLES:
sage: from sage.arith.misc import Moebius sage: p = Moebius().plot() sage: p.ymax() 1.0
- range(start, stop=None, step=None)#
Return the Möbius function evaluated at the given range of values, i.e., the image of the list range(start, stop, step) under the Möbius function.
This is much faster than directly computing all these values with a list comprehension.
EXAMPLES:
sage: v = moebius.range(-10,10); v [1, 0, 0, -1, 1, -1, 0, -1, -1, 1, 0, 1, -1, -1, 0, -1, 1, -1, 0, 0] sage: v == [moebius(n) for n in range(-10,10)] True sage: v = moebius.range(-1000, 2000, 4) sage: v == [moebius(n) for n in range(-1000,2000, 4)] True
- class sage.arith.misc.Sigma#
Bases:
object
Return the sum of the k-th powers of the divisors of n.
INPUT:
n
- integerk
- integer (default: 1)
OUTPUT: integer
EXAMPLES:
sage: sigma(5) 6 sage: sigma(5,2) 26
The sigma function also has a special plotting method.
sage: P = plot(sigma, 1, 100)
This method also works with k-th powers.
sage: P = plot(sigma, 1, 100, k=2)
AUTHORS:
William Stein: original implementation
Craig Citro (2007-06-01): rewrote for huge speedup
- plot(xmin=1, xmax=50, k=1, pointsize=30, rgbcolor=(0, 0, 1), join=True, **kwds)#
Plot the sigma (sum of k-th powers of divisors) function.
INPUT:
xmin
- default: 1xmax
- default: 50k
- default: 1pointsize
- default: 30rgbcolor
- default: (0,0,1)join
- default: True; whether to join the points.**kwds
- passed on
EXAMPLES:
sage: from sage.arith.misc import Sigma sage: p = Sigma().plot() sage: p.ymax() 124.0
- sage.arith.misc.XGCD(a, b)#
Return a triple
(g,s,t)
such that \(g = s\cdot a+t\cdot b = \gcd(a,b)\).Note
One exception is if \(a\) and \(b\) are not in a principal ideal domain (see Wikipedia article Principal_ideal_domain), e.g., they are both polynomials over the integers. Then this function can’t in general return
(g,s,t)
as above, since they need not exist. Instead, over the integers, we first multiply \(g\) by a divisor of the resultant of \(a/g\) and \(b/g\), up to sign.INPUT:
a, b
- integers or more generally, element of a ring for which the xgcd make sense (e.g. a field or univariate polynomials).
OUTPUT:
g, s, t
- such that \(g = s\cdot a + t\cdot b\)
Note
There is no guarantee that the returned cofactors (s and t) are minimal.
EXAMPLES:
sage: xgcd(56, 44) (4, 4, -5) sage: 4*56 + (-5)*44 4 sage: g, a, b = xgcd(5/1, 7/1); g, a, b (1, 3, -2) sage: a*(5/1) + b*(7/1) == g True sage: x = polygen(QQ) sage: xgcd(x^3 - 1, x^2 - 1) (x - 1, 1, -x) sage: K.<g> = NumberField(x^2-3) sage: g.xgcd(g+2) (1, 1/3*g, 0) sage: R.<a,b> = K[] sage: S.<y> = R.fraction_field()[] sage: xgcd(y^2, a*y+b) (1, a^2/b^2, ((-a)/b^2)*y + 1/b) sage: xgcd((b+g)*y^2, (a+g)*y+b) (1, (a^2 + (2*g)*a + 3)/(b^3 + g*b^2), ((-a + (-g))/b^2)*y + 1/b)
Here is an example of a xgcd for two polynomials over the integers, where the linear combination is not the gcd but the gcd multiplied by the resultant:
sage: R.<x> = ZZ[] sage: gcd(2*x*(x-1), x^2) x sage: xgcd(2*x*(x-1), x^2) (2*x, -1, 2) sage: (2*(x-1)).resultant(x) 2
Tests with numpy and gmpy2 types:
sage: from numpy import int8 sage: xgcd(4,int8(8)) (4, 1, 0) sage: xgcd(int8(4),int8(8)) (4, 1, 0) sage: from gmpy2 import mpz sage: xgcd(mpz(4), mpz(8)) (4, 1, 0) sage: xgcd(4, mpz(8)) (4, 1, 0)
- sage.arith.misc.algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False)#
Return an irreducible polynomial of degree at most \(degree\) which is approximately satisfied by the number \(z\).
You can specify the number of known bits or digits of \(z\) with
known_bits=k
orknown_digits=k
. PARI is then told to compute the result using \(0.8k\) of these bits/digits. Or, you can specify the precision to use directly withuse_bits=k
oruse_digits=k
. If none of these are specified, then the precision is taken from the input value.A height bound may be specified to indicate the maximum coefficient size of the returned polynomial; if a sufficiently small polynomial is not found, then
None
will be returned. Ifproof=True
then the result is returned only if it can be proved correct (i.e. the only possible minimal polynomial satisfying the height bound, or no such polynomial exists). Otherwise aValueError
is raised indicating that higher precision is required.ALGORITHM: Uses LLL for real/complex inputs, PARI C-library
algdep
command otherwise.Note that
algebraic_dependency
is a synonym foralgdep
.INPUT:
z
- real, complex, or \(p\)-adic numberdegree
- an integerheight_bound
- an integer (default:None
) specifying the maximumcoefficient size for the returned polynomial
proof
- a boolean (default:False
), requires height_bound to be set
EXAMPLES:
sage: algdep(1.888888888888888, 1) 9*x - 17 sage: algdep(0.12121212121212,1) 33*x - 4 sage: algdep(sqrt(2),2) x^2 - 2
This example involves a complex number:
sage: z = (1/2)*(1 + RDF(sqrt(3)) *CC.0); z 0.500000000000000 + 0.866025403784439*I sage: algdep(z, 6) x^2 - x + 1
This example involves a \(p\)-adic number:
sage: K = Qp(3, print_mode = 'series') sage: a = K(7/19); a 1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20) sage: algdep(a, 1) 19*x - 7
These examples show the importance of proper precision control. We compute a 200-bit approximation to \(sqrt(2)\) which is wrong in the 33’rd bit:
sage: z = sqrt(RealField(200)(2)) + (1/2)^33 sage: p = algdep(z, 4); p 227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088 sage: factor(p) 227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088 sage: algdep(z, 4, known_bits=32) x^2 - 2 sage: algdep(z, 4, known_digits=10) x^2 - 2 sage: algdep(z, 4, use_bits=25) x^2 - 2 sage: algdep(z, 4, use_digits=8) x^2 - 2
Using the
height_bound
andproof
parameters, we can see that \(pi\) is not the root of an integer polynomial of degree at most 5 and coefficients bounded above by 10:sage: algdep(pi.n(), 5, height_bound=10, proof=True) is None True
For stronger results, we need more precision:
sage: algdep(pi.n(), 5, height_bound=100, proof=True) is None Traceback (most recent call last): ... ValueError: insufficient precision for non-existence proof sage: algdep(pi.n(200), 5, height_bound=100, proof=True) is None True sage: algdep(pi.n(), 10, height_bound=10, proof=True) is None Traceback (most recent call last): ... ValueError: insufficient precision for non-existence proof sage: algdep(pi.n(200), 10, height_bound=10, proof=True) is None True
We can also use
proof=True
to get positive results:sage: a = sqrt(2) + sqrt(3) + sqrt(5) sage: algdep(a.n(), 8, height_bound=1000, proof=True) Traceback (most recent call last): ... ValueError: insufficient precision for uniqueness proof sage: f = algdep(a.n(1000), 8, height_bound=1000, proof=True); f x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576 sage: f(a).expand() 0
- sage.arith.misc.algebraic_dependency(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False)#
Return an irreducible polynomial of degree at most \(degree\) which is approximately satisfied by the number \(z\).
You can specify the number of known bits or digits of \(z\) with
known_bits=k
orknown_digits=k
. PARI is then told to compute the result using \(0.8k\) of these bits/digits. Or, you can specify the precision to use directly withuse_bits=k
oruse_digits=k
. If none of these are specified, then the precision is taken from the input value.A height bound may be specified to indicate the maximum coefficient size of the returned polynomial; if a sufficiently small polynomial is not found, then
None
will be returned. Ifproof=True
then the result is returned only if it can be proved correct (i.e. the only possible minimal polynomial satisfying the height bound, or no such polynomial exists). Otherwise aValueError
is raised indicating that higher precision is required.ALGORITHM: Uses LLL for real/complex inputs, PARI C-library
algdep
command otherwise.Note that
algebraic_dependency
is a synonym foralgdep
.INPUT:
z
- real, complex, or \(p\)-adic numberdegree
- an integerheight_bound
- an integer (default:None
) specifying the maximumcoefficient size for the returned polynomial
proof
- a boolean (default:False
), requires height_bound to be set
EXAMPLES:
sage: algdep(1.888888888888888, 1) 9*x - 17 sage: algdep(0.12121212121212,1) 33*x - 4 sage: algdep(sqrt(2),2) x^2 - 2
This example involves a complex number:
sage: z = (1/2)*(1 + RDF(sqrt(3)) *CC.0); z 0.500000000000000 + 0.866025403784439*I sage: algdep(z, 6) x^2 - x + 1
This example involves a \(p\)-adic number:
sage: K = Qp(3, print_mode = 'series') sage: a = K(7/19); a 1 + 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^8 + 2*3^9 + 3^11 + 3^12 + 2*3^15 + 2*3^16 + 3^17 + 2*3^19 + O(3^20) sage: algdep(a, 1) 19*x - 7
These examples show the importance of proper precision control. We compute a 200-bit approximation to \(sqrt(2)\) which is wrong in the 33’rd bit:
sage: z = sqrt(RealField(200)(2)) + (1/2)^33 sage: p = algdep(z, 4); p 227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088 sage: factor(p) 227004321085*x^4 - 216947902586*x^3 - 99411220986*x^2 + 82234881648*x - 211871195088 sage: algdep(z, 4, known_bits=32) x^2 - 2 sage: algdep(z, 4, known_digits=10) x^2 - 2 sage: algdep(z, 4, use_bits=25) x^2 - 2 sage: algdep(z, 4, use_digits=8) x^2 - 2
Using the
height_bound
andproof
parameters, we can see that \(pi\) is not the root of an integer polynomial of degree at most 5 and coefficients bounded above by 10:sage: algdep(pi.n(), 5, height_bound=10, proof=True) is None True
For stronger results, we need more precision:
sage: algdep(pi.n(), 5, height_bound=100, proof=True) is None Traceback (most recent call last): ... ValueError: insufficient precision for non-existence proof sage: algdep(pi.n(200), 5, height_bound=100, proof=True) is None True sage: algdep(pi.n(), 10, height_bound=10, proof=True) is None Traceback (most recent call last): ... ValueError: insufficient precision for non-existence proof sage: algdep(pi.n(200), 10, height_bound=10, proof=True) is None True
We can also use
proof=True
to get positive results:sage: a = sqrt(2) + sqrt(3) + sqrt(5) sage: algdep(a.n(), 8, height_bound=1000, proof=True) Traceback (most recent call last): ... ValueError: insufficient precision for uniqueness proof sage: f = algdep(a.n(1000), 8, height_bound=1000, proof=True); f x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576 sage: f(a).expand() 0
- sage.arith.misc.bernoulli(n, algorithm='default', num_threads=1)#
Return the n-th Bernoulli number, as a rational number.
INPUT:
n
- an integeralgorithm
:'default'
– use ‘flint’ for n <= 20000, then ‘arb’ for n <= 300000 and ‘bernmm’ for larger values (this is just a heuristic, and not guaranteed to be optimal on all hardware)'arb'
– use the arb library'flint'
– use the FLINT library'pari'
– use the PARI C library'gap'
– use GAP'gp'
– use PARI/GP interpreter'magma'
– use MAGMA (optional)'bernmm'
– use bernmm package (a multimodular algorithm)
num_threads
- positive integer, number of threads to use (only used for bernmm algorithm)
EXAMPLES:
sage: bernoulli(12) -691/2730 sage: bernoulli(50) 495057205241079648212477525/66
We demonstrate each of the alternative algorithms:
sage: bernoulli(12, algorithm='arb') -691/2730 sage: bernoulli(12, algorithm='flint') -691/2730 sage: bernoulli(12, algorithm='gap') -691/2730 sage: bernoulli(12, algorithm='gp') -691/2730 sage: bernoulli(12, algorithm='magma') # optional - magma -691/2730 sage: bernoulli(12, algorithm='pari') -691/2730 sage: bernoulli(12, algorithm='bernmm') -691/2730 sage: bernoulli(12, algorithm='bernmm', num_threads=4) -691/2730
AUTHOR:
David Joyner and William Stein
- sage.arith.misc.binomial(x, m, **kwds)#
Return the binomial coefficient
\[\binom{x}{m} = x (x-1) \cdots (x-m+1) / m!\]which is defined for \(m \in \ZZ\) and any \(x\). We extend this definition to include cases when \(x-m\) is an integer but \(m\) is not by
\[\binom{x}{m} = \binom{x}{x-m}\]If \(m < 0\), return \(0\).
INPUT:
x
,m
- numbers or symbolic expressions. Eitherm
orx-m
must be an integer.
OUTPUT: number or symbolic expression (if input is symbolic)
EXAMPLES:
sage: from sage.arith.misc import binomial sage: binomial(5,2) 10 sage: binomial(2,0) 1 sage: binomial(1/2, 0) 1 sage: binomial(3,-1) 0 sage: binomial(20,10) 184756 sage: binomial(-2, 5) -6 sage: binomial(-5, -2) 0 sage: binomial(RealField()('2.5'), 2) 1.87500000000000 sage: n=var('n'); binomial(n,2) 1/2*(n - 1)*n sage: n=var('n'); binomial(n,n) 1 sage: n=var('n'); binomial(n,n-1) n sage: binomial(2^100, 2^100) 1 sage: x = polygen(ZZ) sage: binomial(x, 3) 1/6*x^3 - 1/2*x^2 + 1/3*x sage: binomial(x, x-3) 1/6*x^3 - 1/2*x^2 + 1/3*x
If \(x \in \ZZ\), there is an optional ‘algorithm’ parameter, which can be ‘gmp’ (faster for small values; alias: ‘mpir’) or ‘pari’ (faster for large values):
sage: a = binomial(100, 45, algorithm='gmp') sage: b = binomial(100, 45, algorithm='pari') sage: a == b True
- sage.arith.misc.binomial_coefficients(n)#
Return a dictionary containing pairs \(\{(k_1,k_2) : C_{k,n}\}\) where \(C_{k_n}\) are binomial coefficients and \(n = k_1 + k_2\).
INPUT:
n
- an integer
OUTPUT: dict
EXAMPLES:
sage: sorted(binomial_coefficients(3).items()) [((0, 3), 1), ((1, 2), 3), ((2, 1), 3), ((3, 0), 1)]
Notice the coefficients above are the same as below:
sage: R.<x,y> = QQ[] sage: (x+y)^3 x^3 + 3*x^2*y + 3*x*y^2 + y^3
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: sorted(binomial_coefficients(int8(3)).items()) [((0, 3), 1), ((1, 2), 3), ((2, 1), 3), ((3, 0), 1)] sage: from gmpy2 import mpz sage: sorted(binomial_coefficients(mpz(3)).items()) [((0, 3), 1), ((1, 2), 3), ((2, 1), 3), ((3, 0), 1)]
AUTHORS:
Fredrik Johansson
- sage.arith.misc.continuant(v, n=None)#
Function returns the continuant of the sequence \(v\) (list or tuple).
Definition: see Graham, Knuth and Patashnik, Concrete Mathematics, section 6.7: Continuants. The continuant is defined by
\(K_0() = 1\)
\(K_1(x_1) = x_1\)
\(K_n(x_1, \cdots, x_n) = K_{n-1}(x_n, \cdots x_{n-1})x_n + K_{n-2}(x_1, \cdots, x_{n-2})\)
If
n = None
orn > len(v)
the defaultn = len(v)
is used.INPUT:
v
- list or tuple of elements of a ringn
- optional integer
OUTPUT: element of ring (integer, polynomial, etcetera).
EXAMPLES:
sage: continuant([1,2,3]) 10 sage: p = continuant([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10]) sage: q = continuant([1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10]) sage: p/q 517656/190435 sage: continued_fraction([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10]).convergent(14) 517656/190435 sage: x = PolynomialRing(RationalField(),'x',5).gens() sage: continuant(x) x0*x1*x2*x3*x4 + x0*x1*x2 + x0*x1*x4 + x0*x3*x4 + x2*x3*x4 + x0 + x2 + x4 sage: continuant(x, 3) x0*x1*x2 + x0 + x2 sage: continuant(x,2) x0*x1 + 1
We verify the identity
\[K_n(z,z,\cdots,z) = \sum_{k=0}^n \binom{n-k}{k} z^{n-2k}\]for \(n = 6\) using polynomial arithmetic:
sage: z = QQ['z'].0 sage: continuant((z,z,z,z,z,z,z,z,z,z,z,z,z,z,z),6) z^6 + 5*z^4 + 6*z^2 + 1 sage: continuant(9) Traceback (most recent call last): ... TypeError: object of type 'sage.rings.integer.Integer' has no len()
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: continuant([int8(1),int8(2),int8(3)]) 10 sage: from gmpy2 import mpz sage: continuant([mpz(1),mpz(2),mpz(3)]) mpz(10)
AUTHORS:
Jaap Spies (2007-02-06)
- sage.arith.misc.crt(a, b, m=None, n=None)#
Return a solution to a Chinese Remainder Theorem problem.
INPUT:
a
,b
- two residues (elements of some ring for which extended gcd is available), or two lists, one of residues and one of moduli.m
,n
- (default:None
) two moduli, orNone
.
OUTPUT:
If
m
,n
are notNone
, returns a solution \(x\) to the simultaneous congruences \(x\equiv a \bmod m\) and \(x\equiv b \bmod n\), if one exists. By the Chinese Remainder Theorem, a solution to the simultaneous congruences exists if and only if \(a\equiv b\pmod{\gcd(m,n)}\). The solution \(x\) is only well-defined modulo \(\text{lcm}(m,n)\).If
a
andb
are lists, returns a simultaneous solution to the congruences \(x\equiv a_i\pmod{b_i}\), if one exists.See also
EXAMPLES:
Using
crt
by giving it pairs of residues and moduli:sage: crt(2, 1, 3, 5) 11 sage: crt(13, 20, 100, 301) 28013 sage: crt([2, 1], [3, 5]) 11 sage: crt([13, 20], [100, 301]) 28013
You can also use upper case:
sage: c = CRT(2,3, 3, 5); c 8 sage: c % 3 == 2 True sage: c % 5 == 3 True
Note that this also works for polynomial rings:
sage: K.<a> = NumberField(x^3 - 7) sage: R.<y> = K[] sage: f = y^2 + 3 sage: g = y^3 - 5 sage: CRT(1,3,f,g) -3/26*y^4 + 5/26*y^3 + 15/26*y + 53/26 sage: CRT(1,a,f,g) (-3/52*a + 3/52)*y^4 + (5/52*a - 5/52)*y^3 + (15/52*a - 15/52)*y + 27/52*a + 25/52
You can also do this for any number of moduli:
sage: K.<a> = NumberField(x^3 - 7) sage: R.<x> = K[] sage: CRT([], []) 0 sage: CRT([a], [x]) a sage: f = x^2 + 3 sage: g = x^3 - 5 sage: h = x^5 + x^2 - 9 sage: k = CRT([1, a, 3], [f, g, h]); k (127/26988*a - 5807/386828)*x^9 + (45/8996*a - 33677/1160484)*x^8 + (2/173*a - 6/173)*x^7 + (133/6747*a - 5373/96707)*x^6 + (-6/2249*a + 18584/290121)*x^5 + (-277/8996*a + 38847/386828)*x^4 + (-135/4498*a + 42673/193414)*x^3 + (-1005/8996*a + 470245/1160484)*x^2 + (-1215/8996*a + 141165/386828)*x + 621/8996*a + 836445/386828 sage: k.mod(f) 1 sage: k.mod(g) a sage: k.mod(h) 3
If the moduli are not coprime, a solution may not exist:
sage: crt(4,8,8,12) 20 sage: crt(4,6,8,12) Traceback (most recent call last): ... ValueError: no solution to crt problem since gcd(8,12) does not divide 4-6 sage: x = polygen(QQ) sage: crt(2,3,x-1,x+1) -1/2*x + 5/2 sage: crt(2,x,x^2-1,x^2+1) -1/2*x^3 + x^2 + 1/2*x + 1 sage: crt(2,x,x^2-1,x^3-1) Traceback (most recent call last): ... ValueError: no solution to crt problem since gcd(x^2 - 1,x^3 - 1) does not divide 2-x sage: crt(int(2), int(3), int(7), int(11)) 58
crt also work with numpy and gmpy2 numbers:
sage: import numpy sage: crt(numpy.int8(2), numpy.int8(3), numpy.int8(7), numpy.int8(11)) 58 sage: from gmpy2 import mpz sage: crt(mpz(2), mpz(3), mpz(7), mpz(11)) 58 sage: crt(mpz(2), 3, mpz(7), numpy.int8(11)) 58
- sage.arith.misc.dedekind_psi(N)#
Return the value of the Dedekind psi function at
N
.INPUT:
N
– a positive integer
OUTPUT:
an integer
The Dedekind psi function is the multiplicative function defined by
\[\psi(n) = n \prod_{p|n, p prime} (1 + 1/p).\]See Wikipedia article Dedekind_psi_function and OEIS sequence A001615.
EXAMPLES:
sage: from sage.arith.misc import dedekind_psi sage: [dedekind_psi(d) for d in range(1, 12)] [1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12]
- sage.arith.misc.dedekind_sum(p, q, algorithm='default')#
Return the Dedekind sum \(s(p,q)\) defined for integers \(p\), \(q\) as
\[s(p,q) = \sum_{i=0}^{q-1} \left(\!\left(\frac{i}{q}\right)\!\right) \left(\!\left(\frac{pi}{q}\right)\!\right)\]where
\[\begin{split}((x))=\begin{cases} x-\lfloor x \rfloor - \frac{1}{2} &\mbox{if } x \in \QQ \setminus \ZZ \\ 0 & \mbox{if } x \in \ZZ. \end{cases}\end{split}\]Warning
Caution is required as the Dedekind sum sometimes depends on the algorithm or is left undefined when \(p\) and \(q\) are not coprime.
INPUT:
p
,q
– integersalgorithm
– must be one of the following'default'
- (default) use FLINT'flint'
- use FLINT'pari'
- use PARI (gives different results if \(p\) and \(q\) are not coprime)
OUTPUT: a rational number
EXAMPLES:
Several small values:
sage: for q in range(10): print([dedekind_sum(p,q) for p in range(q+1)]) [0] [0, 0] [0, 0, 0] [0, 1/18, -1/18, 0] [0, 1/8, 0, -1/8, 0] [0, 1/5, 0, 0, -1/5, 0] [0, 5/18, 1/18, 0, -1/18, -5/18, 0] [0, 5/14, 1/14, -1/14, 1/14, -1/14, -5/14, 0] [0, 7/16, 1/8, 1/16, 0, -1/16, -1/8, -7/16, 0] [0, 14/27, 4/27, 1/18, -4/27, 4/27, -1/18, -4/27, -14/27, 0]
Check relations for restricted arguments:
sage: q = 23; dedekind_sum(1, q); (q-1)*(q-2)/(12*q) 77/46 77/46 sage: p, q = 100, 723 # must be coprime sage: dedekind_sum(p, q) + dedekind_sum(q, p) 31583/86760 sage: -1/4 + (p/q + q/p + 1/(p*q))/12 31583/86760
We check that evaluation works with large input:
sage: dedekind_sum(3^54 - 1, 2^93 + 1) 459340694971839990630374299870/29710560942849126597578981379 sage: dedekind_sum(3^54 - 1, 2^93 + 1, algorithm='pari') 459340694971839990630374299870/29710560942849126597578981379
We check consistency of the results:
sage: dedekind_sum(5, 7, algorithm='default') -1/14 sage: dedekind_sum(5, 7, algorithm='flint') -1/14 sage: dedekind_sum(5, 7, algorithm='pari') -1/14 sage: dedekind_sum(6, 8, algorithm='default') -1/8 sage: dedekind_sum(6, 8, algorithm='flint') -1/8 sage: dedekind_sum(6, 8, algorithm='pari') -1/8
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: dedekind_sum(int8(5), int8(7), algorithm='default') -1/14 sage: from gmpy2 import mpz sage: dedekind_sum(mpz(5), mpz(7), algorithm='default') -1/14
REFERENCES:
- sage.arith.misc.differences(lis, n=1)#
Return the \(n\) successive differences of the elements in
lis
.EXAMPLES:
sage: differences(prime_range(50)) [1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4] sage: differences([i^2 for i in range(1,11)]) [3, 5, 7, 9, 11, 13, 15, 17, 19] sage: differences([i^3 + 3*i for i in range(1,21)]) [10, 22, 40, 64, 94, 130, 172, 220, 274, 334, 400, 472, 550, 634, 724, 820, 922, 1030, 1144] sage: differences([i^3 - i^2 for i in range(1,21)], 2) [10, 16, 22, 28, 34, 40, 46, 52, 58, 64, 70, 76, 82, 88, 94, 100, 106, 112] sage: differences([p - i^2 for i, p in enumerate(prime_range(50))], 3) [-1, 2, -4, 4, -4, 4, 0, -6, 8, -6, 0, 4]
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: differences([int8(1),int8(4),int8(6),int8(19)]) [3, 2, 13] sage: from gmpy2 import mpz sage: differences([mpz(1),mpz(4),mpz(6),mpz(19)]) [mpz(3), mpz(2), mpz(13)]
AUTHORS:
Timothy Clemans (2008-03-09)
- sage.arith.misc.divisors(n)#
Return the list of all divisors (up to units) of this element of a unique factorization domain.
For an integer, the list of all positive integer divisors of this integer, sorted in increasing order, is returned.
INPUT:
n
- the element
EXAMPLES:
Divisors of integers:
sage: divisors(-3) [1, 3] sage: divisors(6) [1, 2, 3, 6] sage: divisors(28) [1, 2, 4, 7, 14, 28] sage: divisors(2^5) [1, 2, 4, 8, 16, 32] sage: divisors(100) [1, 2, 4, 5, 10, 20, 25, 50, 100] sage: divisors(1) [1] sage: divisors(0) Traceback (most recent call last): ... ValueError: n must be nonzero sage: divisors(2^3 * 3^2 * 17) [1, 2, 3, 4, 6, 8, 9, 12, 17, 18, 24, 34, 36, 51, 68, 72, 102, 136, 153, 204, 306, 408, 612, 1224]
This function works whenever one has unique factorization:
sage: K.<a> = QuadraticField(7) sage: divisors(K.ideal(7)) [Fractional ideal (1), Fractional ideal (-a), Fractional ideal (7)] sage: divisors(K.ideal(3)) [Fractional ideal (1), Fractional ideal (3), Fractional ideal (-a + 2), Fractional ideal (-a - 2)] sage: divisors(K.ideal(35)) [Fractional ideal (1), Fractional ideal (5), Fractional ideal (-a), Fractional ideal (7), Fractional ideal (-5*a), Fractional ideal (35)]
- sage.arith.misc.eratosthenes(n)#
Return a list of the primes \(\leq n\).
This is extremely slow and is for educational purposes only.
INPUT:
n
- a positive integer
OUTPUT:
a list of primes less than or equal to n.
EXAMPLES:
sage: eratosthenes(3) [2, 3] sage: eratosthenes(50) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47] sage: len(eratosthenes(100)) 25 sage: eratosthenes(213) == prime_range(213) True
- sage.arith.misc.factor(n, proof=None, int_=False, algorithm='pari', verbose=0, **kwds)#
Return the factorization of
n
. The result depends on the type ofn
.If
n
is an integer, returns the factorization as an object of typeFactorization
.If n is not an integer,
n.factor(proof=proof, **kwds)
gets called. Seen.factor??
for more documentation in this case.Warning
This means that applying
factor
to an integer result of a symbolic computation will not factor the integer, because it is considered as an element of a larger symbolic ring.EXAMPLES:
sage: f(n)=n^2 sage: is_prime(f(3)) False sage: factor(f(3)) 9
INPUT:
n
- an nonzero integerproof
- bool or None (default: None)int_
- bool (default: False) whether to return answers as Python intsalgorithm
- string'pari'
- (default) use the PARI c library'kash'
- use KASH computer algebra system (requires that kash be installed)'magma'
- use Magma (requires magma be installed)
verbose
- integer (default: 0); PARI’s debug variable is set to this; e.g., set to 4 or 8 to see lots of output during factorization.
OUTPUT:
factorization of n
The qsieve and ecm commands give access to highly optimized implementations of algorithms for doing certain integer factorization problems. These implementations are not used by the generic factor command, which currently just calls PARI (note that PARI also implements sieve and ecm algorithms, but they are not as optimized). Thus you might consider using them instead for certain numbers.
The factorization returned is an element of the class
Factorization
; see Factorization?? for more details, and examples below for usage. A Factorization contains both the unit factor (+1 or -1) and a sorted list of (prime, exponent) pairs.The factorization displays in pretty-print format but it is easy to obtain access to the (prime,exponent) pairs and the unit, to recover the number from its factorization, and even to multiply two factorizations. See examples below.
EXAMPLES:
sage: factor(500) 2^2 * 5^3 sage: factor(-20) -1 * 2^2 * 5 sage: f=factor(-20) sage: list(f) [(2, 2), (5, 1)] sage: f.unit() -1 sage: f.value() -20 sage: factor( -next_prime(10^2) * next_prime(10^7) ) -1 * 101 * 10000019
sage: factor(-500, algorithm='kash') # optional - kash -1 * 2^2 * 5^3
sage: factor(-500, algorithm='magma') # optional - magma -1 * 2^2 * 5^3
sage: factor(0) Traceback (most recent call last): ... ArithmeticError: factorization of 0 is not defined sage: factor(1) 1 sage: factor(-1) -1 sage: factor(2^(2^7)+1) 59649589127497217 * 5704689200685129054721
Sage calls PARI’s factor, which has proof False by default. Sage has a global proof flag, set to True by default (see
sage.structure.proof.proof
, or proof.[tab]). To override the default, call this function with proof=False.sage: factor(3^89-1, proof=False) 2 * 179 * 1611479891519807 * 5042939439565996049162197
sage: factor(2^197 + 1) # long time (2s) 3 * 197002597249 * 1348959352853811313 * 251951573867253012259144010843
Any object which has a factor method can be factored like this:
sage: K.<i> = QuadraticField(-1) sage: factor(122 - 454*i) (-3*i - 2) * (-i - 2)^3 * (i + 1)^3 * (i + 4)
To access the data in a factorization:
sage: f = factor(420); f 2^2 * 3 * 5 * 7 sage: [x for x in f] [(2, 2), (3, 1), (5, 1), (7, 1)] sage: [p for p,e in f] [2, 3, 5, 7] sage: [e for p,e in f] [2, 1, 1, 1] sage: [p^e for p,e in f] [4, 3, 5, 7]
We can factor Python, numpy and gmpy2 numbers:
sage: factor(math.pi) 3.141592653589793 sage: import numpy sage: factor(numpy.int8(30)) 2 * 3 * 5 sage: import gmpy2 sage: factor(gmpy2.mpz(30)) 2 * 3 * 5
- sage.arith.misc.factorial(n, algorithm='gmp')#
Compute the factorial of \(n\), which is the product \(1\cdot 2\cdot 3 \cdots (n-1)\cdot n\).
INPUT:
n
- an integeralgorithm
- string (default: ‘gmp’):'gmp'
- use the GMP C-library factorial function'pari'
- use PARI’s factorial function
OUTPUT: an integer
EXAMPLES:
sage: from sage.arith.misc import factorial sage: factorial(0) 1 sage: factorial(4) 24 sage: factorial(10) 3628800 sage: factorial(1) == factorial(0) True sage: factorial(6) == 6*5*4*3*2 True sage: factorial(1) == factorial(0) True sage: factorial(71) == 71* factorial(70) True sage: factorial(-32) Traceback (most recent call last): ... ValueError: factorial -- must be nonnegative
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: factorial(int8(4)) 24 sage: from gmpy2 import mpz sage: factorial(mpz(4)) 24
PERFORMANCE: This discussion is valid as of April 2006. All timings below are on a Pentium Core Duo 2Ghz MacBook Pro running Linux with a 2.6.16.1 kernel.
It takes less than a minute to compute the factorial of \(10^7\) using the GMP algorithm, and the factorial of \(10^6\) takes less than 4 seconds.
The GMP algorithm is faster and more memory efficient than the PARI algorithm. E.g., PARI computes \(10^7\) factorial in 100 seconds on the core duo 2Ghz.
For comparison, computation in Magma \(\leq\) 2.12-10 of \(n!\) is best done using
*[1..n]
. It takes 113 seconds to compute the factorial of \(10^7\) and 6 seconds to compute the factorial of \(10^6\). Mathematica V5.2 compute the factorial of \(10^7\) in 136 seconds and the factorial of \(10^6\) in 7 seconds. (Mathematica is notably very efficient at memory usage when doing factorial calculations.)
- sage.arith.misc.falling_factorial(x, a)#
Return the falling factorial \((x)_a\).
The notation in the literature is a mess: often \((x)_a\), but there are many other notations: GKP: Concrete Mathematics uses \(x^{\underline{a}}\).
Definition: for integer \(a \ge 0\) we have \(x(x-1) \cdots (x-a+1)\). In all other cases we use the GAMMA-function: \(\frac {\Gamma(x+1)} {\Gamma(x-a+1)}\).
INPUT:
x
– element of a ringa
– a non-negative integer orx and a
– any numbers
OUTPUT: the falling factorial
See also
EXAMPLES:
sage: falling_factorial(10, 3) 720 sage: falling_factorial(10, RR('3.0')) 720.000000000000 sage: falling_factorial(10, RR('3.3')) 1310.11633396601 sage: falling_factorial(10, 10) 3628800 sage: factorial(10) 3628800 sage: a = falling_factorial(1+I, I); a gamma(I + 2) sage: CC(a) 0.652965496420167 + 0.343065839816545*I sage: falling_factorial(1+I, 4) 4*I + 2 sage: falling_factorial(I, 4) -10 sage: M = MatrixSpace(ZZ, 4, 4) sage: A = M([1,0,1,0,1,0,1,0,1,0,10,10,1,0,1,1]) sage: falling_factorial(A, 2) # A(A - I) [ 1 0 10 10] [ 1 0 10 10] [ 20 0 101 100] [ 2 0 11 10] sage: x = ZZ['x'].0 sage: falling_factorial(x, 4) x^4 - 6*x^3 + 11*x^2 - 6*x
AUTHORS:
Jaap Spies (2006-03-05)
- sage.arith.misc.four_squares(n)#
Write the integer \(n\) as a sum of four integer squares.
INPUT:
n
– an integer
OUTPUT: a tuple \((a,b,c,d)\) of non-negative integers such that \(n = a^2 + b^2 + c^2 + d^2\) with \(a <= b <= c <= d\).
EXAMPLES:
sage: four_squares(3) (0, 1, 1, 1) sage: four_squares(13) (0, 0, 2, 3) sage: four_squares(130) (0, 0, 3, 11) sage: four_squares(1101011011004) (90, 102, 1220, 1049290) sage: four_squares(10^100-1) (155024616290, 2612183768627, 14142135623730950488016887, 99999999999999999999999999999999999999999999999999) sage: for i in range(2^129, 2^129+10000): # long time ....: S = four_squares(i) ....: assert sum(x^2 for x in S) == i
- sage.arith.misc.fundamental_discriminant(D)#
Return the discriminant of the quadratic extension \(K=Q(\sqrt{D})\), i.e. an integer d congruent to either 0 or 1, mod 4, and such that, at most, the only square dividing it is 4.
INPUT:
D
- an integer
OUTPUT:
an integer, the fundamental discriminant
EXAMPLES:
sage: fundamental_discriminant(102) 408 sage: fundamental_discriminant(720) 5 sage: fundamental_discriminant(2) 8
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: fundamental_discriminant(int8(102)) 408 sage: from gmpy2 import mpz sage: fundamental_discriminant(mpz(102)) 408
- sage.arith.misc.gauss_sum(char_value, finite_field)#
Return the Gauss sums for a general finite field.
INPUT:
char_value
– choice of multiplicative character, given by its value on thefinite_field.multiplicative_generator()
finite_field
– a finite field
OUTPUT:
an element of the parent ring of
char_value
, that can be any field containing enough roots of unity, for example theUniversalCyclotomicField
,QQbar
orComplexField
For a finite field \(F\) of characteristic \(p\), the Gauss sum associated to a multiplicative character \(\chi\) (with values in a ring \(K\)) is defined as
\[\sum_{x \in F^{\times}} \chi(x) \zeta_p^{\operatorname{Tr} x},\]where \(\zeta_p \in K\) is a primitive root of unity of order \(p\) and Tr is the trace map from \(F\) to its prime field \(\GF{p}\).
For more info on Gauss sums, see Wikipedia article Gauss_sum.
Todo
Implement general Gauss sums for an arbitrary pair
(multiplicative_character, additive_character)
EXAMPLES:
sage: from sage.arith.misc import gauss_sum sage: F = GF(5); q = 5 sage: zq = UniversalCyclotomicField().zeta(q-1) sage: L = [gauss_sum(zq**i,F) for i in range(5)]; L [-1, E(20)^4 + E(20)^13 - E(20)^16 - E(20)^17, E(5) - E(5)^2 - E(5)^3 + E(5)^4, E(20)^4 - E(20)^13 - E(20)^16 + E(20)^17, -1] sage: [g*g.conjugate() for g in L] [1, 5, 5, 5, 1] sage: F = GF(11**2); q = 11**2 sage: zq = UniversalCyclotomicField().zeta(q-1) sage: g = gauss_sum(zq**4,F) sage: g*g.conjugate() 121
See also
sage.rings.padics.misc.gauss_sum()
for a \(p\)-adic versionsage.modular.dirichlet.DirichletCharacter.gauss_sum()
for prime finite fieldssage.modular.dirichlet.DirichletCharacter.gauss_sum_numerical()
for prime finite fields
- sage.arith.misc.gcd(a, b=None, **kwargs)#
Return the greatest common divisor of
a
andb
.If
a
is a list andb
is omitted, return instead the greatest common divisor of all elements ofa
.INPUT:
a,b
– two elements of a ring with gcd ora
– a list or tuple of elements of a ring with gcd
Additional keyword arguments are passed to the respectively called methods.
OUTPUT:
The given elements are first coerced into a common parent. Then, their greatest common divisor in that common parent is returned.
EXAMPLES:
sage: GCD(97,100) 1 sage: GCD(97*10^15, 19^20*97^2) 97 sage: GCD(2/3, 4/5) 2/15 sage: GCD([2,4,6,8]) 2 sage: GCD(srange(0,10000,10)) # fast !! 10
Note that to take the gcd of \(n\) elements for \(n \not= 2\) you must put the elements into a list by enclosing them in
[..]
. Before trac ticket #4988 the following wrongly returned 3 since the third parameter was just ignored:sage: gcd(3, 6, 2) Traceback (most recent call last): ... TypeError: ...gcd() takes ... sage: gcd([3, 6, 2]) 1
Similarly, giving just one element (which is not a list) gives an error:
sage: gcd(3) Traceback (most recent call last): ... TypeError: 'sage.rings.integer.Integer' object is not iterable
By convention, the gcd of the empty list is (the integer) 0:
sage: gcd([]) 0 sage: type(gcd([])) <class 'sage.rings.integer.Integer'>
- sage.arith.misc.get_gcd(order)#
Return the fastest gcd function for integers of size no larger than order.
EXAMPLES:
sage: sage.arith.misc.get_gcd(4000) <built-in method gcd_int of sage.rings.fast_arith.arith_int object at ...> sage: sage.arith.misc.get_gcd(400000) <built-in method gcd_longlong of sage.rings.fast_arith.arith_llong object at ...> sage: sage.arith.misc.get_gcd(4000000000) <function gcd at ...>
- sage.arith.misc.get_inverse_mod(order)#
Return the fastest inverse_mod function for integers of size no larger than order.
EXAMPLES:
sage: sage.arith.misc.get_inverse_mod(6000) <built-in method inverse_mod_int of sage.rings.fast_arith.arith_int object at ...> sage: sage.arith.misc.get_inverse_mod(600000) <built-in method inverse_mod_longlong of sage.rings.fast_arith.arith_llong object at ...> sage: sage.arith.misc.get_inverse_mod(6000000000) <function inverse_mod at ...>
- sage.arith.misc.hilbert_conductor(a, b)#
Return the product of all (finite) primes where the Hilbert symbol is -1.
This is the (reduced) discriminant of the quaternion algebra \((a,b)\) over \(\QQ\).
INPUT:
a
,b
– integers
OUTPUT:
squarefree positive integer
EXAMPLES:
sage: hilbert_conductor(-1, -1) 2 sage: hilbert_conductor(-1, -11) 11 sage: hilbert_conductor(-2, -5) 5 sage: hilbert_conductor(-3, -17) 17
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: hilbert_conductor(int8(-3), int8(-17)) 17 sage: from gmpy2 import mpz sage: hilbert_conductor(mpz(-3), mpz(-17)) 17
AUTHOR:
Gonzalo Tornaria (2009-03-02)
- sage.arith.misc.hilbert_conductor_inverse(d)#
Finds a pair of integers \((a,b)\) such that
hilbert_conductor(a,b) == d
.The quaternion algebra \((a,b)\) over \(\QQ\) will then have (reduced) discriminant \(d\).
INPUT:
d
– square-free positive integer
OUTPUT: pair of integers
EXAMPLES:
sage: hilbert_conductor_inverse(2) (-1, -1) sage: hilbert_conductor_inverse(3) (-1, -3) sage: hilbert_conductor_inverse(6) (-1, 3) sage: hilbert_conductor_inverse(30) (-3, -10) sage: hilbert_conductor_inverse(4) Traceback (most recent call last): ... ValueError: d needs to be squarefree sage: hilbert_conductor_inverse(-1) Traceback (most recent call last): ... ValueError: d needs to be positive
AUTHOR:
Gonzalo Tornaria (2009-03-02)
- sage.arith.misc.hilbert_symbol(a, b, p, algorithm='pari')#
Return 1 if \(ax^2 + by^2\) \(p\)-adically represents a nonzero square, otherwise returns \(-1\). If either a or b is 0, returns 0.
INPUT:
a, b
- integersp
- integer; either prime or -1 (which represents the archimedean place)algorithm
- string'pari'
- (default) use the PARI C library'direct'
- use a Python implementation'all'
- use both PARI and direct and check that the results agree, then return the common answer
OUTPUT: integer (0, -1, or 1)
EXAMPLES:
sage: hilbert_symbol (-1, -1, -1, algorithm='all') -1 sage: hilbert_symbol (2,3, 5, algorithm='all') 1 sage: hilbert_symbol (4, 3, 5, algorithm='all') 1 sage: hilbert_symbol (0, 3, 5, algorithm='all') 0 sage: hilbert_symbol (-1, -1, 2, algorithm='all') -1 sage: hilbert_symbol (1, -1, 2, algorithm='all') 1 sage: hilbert_symbol (3, -1, 2, algorithm='all') -1 sage: hilbert_symbol(QQ(-1)/QQ(4), -1, 2) == -1 True sage: hilbert_symbol(QQ(-1)/QQ(4), -1, 3) == 1 True
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: hilbert_symbol(int8(2),int8(3),int8(5),algorithm='all') 1 sage: from gmpy2 import mpz sage: hilbert_symbol(mpz(2),mpz(3),mpz(5),algorithm='all') 1
AUTHORS:
William Stein and David Kohel (2006-01-05)
- sage.arith.misc.integer_ceil(x)#
Return the ceiling of x.
EXAMPLES:
sage: integer_ceil(5.4) 6 sage: integer_ceil(x) Traceback (most recent call last): ... NotImplementedError: computation of ceil of x not implemented
Tests with numpy and gmpy2 numbers:
sage: from numpy import float32 sage: integer_ceil(float32(5.4)) 6 sage: from gmpy2 import mpfr sage: integer_ceil(mpfr(5.4)) 6
- sage.arith.misc.integer_floor(x)#
Return the largest integer \(\leq x\).
INPUT:
x
- an object that has a floor method or is coercible to int
OUTPUT: an Integer
EXAMPLES:
sage: integer_floor(5.4) 5 sage: integer_floor(float(5.4)) 5 sage: integer_floor(-5/2) -3 sage: integer_floor(RDF(-5/2)) -3 sage: integer_floor(x) Traceback (most recent call last): ... NotImplementedError: computation of floor of x not implemented
Tests with numpy and gmpy2 numbers:
sage: from numpy import float32 sage: integer_floor(float32(5.4)) 5 sage: from gmpy2 import mpfr sage: integer_floor(mpfr(5.4)) 5
- sage.arith.misc.integer_trunc(i)#
Truncate to the integer closer to zero
EXAMPLES:
sage: from sage.arith.misc import integer_trunc as trunc sage: trunc(-3/2), trunc(-1), trunc(-1/2), trunc(0), trunc(1/2), trunc(1), trunc(3/2) (-1, -1, 0, 0, 0, 1, 1) sage: isinstance(trunc(3/2), Integer) True
- sage.arith.misc.inverse_mod(a, m)#
The inverse of the ring element a modulo m.
If no special inverse_mod is defined for the elements, it tries to coerce them into integers and perform the inversion there
sage: inverse_mod(7,1) 0 sage: inverse_mod(5,14) 3 sage: inverse_mod(3,-5) 2
Tests with numpy and mpz numbers:
sage: from numpy import int8 sage: inverse_mod(int8(5),int8(14)) 3 sage: from gmpy2 import mpz sage: inverse_mod(mpz(5),mpz(14)) 3
- sage.arith.misc.is_power_of_two(n)#
Return whether
n
is a power of 2.INPUT:
n
– integer
OUTPUT:
boolean
EXAMPLES:
sage: is_power_of_two(1024) True sage: is_power_of_two(1) True sage: is_power_of_two(24) False sage: is_power_of_two(0) False sage: is_power_of_two(-4) False
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: is_power_of_two(int8(16)) True sage: is_power_of_two(int8(24)) False sage: from gmpy2 import mpz sage: is_power_of_two(mpz(16)) True sage: is_power_of_two(mpz(24)) False
- sage.arith.misc.is_prime(n)#
Determine whether \(n\) is a prime element of its parent ring.
INPUT:
n
– the object for which to determine primality
Exceptional special cases:
For integers, determine whether \(n\) is a positive prime.
For number fields except \(\QQ\), determine whether \(n\) is a prime element of the maximal order.
ALGORITHM:
For integers, this function uses a provable primality test or a strong pseudo-primality test depending on the global
arithmetic proof flag
.EXAMPLES:
sage: is_prime(389) True sage: is_prime(2000) False sage: is_prime(2) True sage: is_prime(-1) False sage: is_prime(1) False sage: is_prime(-2) False
sage: a = 2**2048 + 981 sage: is_prime(a) # not tested - takes ~ 1min sage: proof.arithmetic(False) sage: is_prime(a) # instantaneous! True sage: proof.arithmetic(True)
- sage.arith.misc.is_prime_power(n, get_data=False)#
Test whether
n
is a positive power of a prime numberThis function simply calls the method
Integer.is_prime_power()
of Integers.INPUT:
n
– an integerget_data
– if set toTrue
, return a pair(p,k)
such that this integer equalsp^k
instead ofTrue
or(self,0)
instead ofFalse
EXAMPLES:
sage: is_prime_power(389) True sage: is_prime_power(2000) False sage: is_prime_power(2) True sage: is_prime_power(1024) True sage: is_prime_power(1024, get_data=True) (2, 10)
The same results can be obtained with:
sage: 389.is_prime_power() True sage: 2000.is_prime_power() False sage: 2.is_prime_power() True sage: 1024.is_prime_power() True sage: 1024.is_prime_power(get_data=True) (2, 10)
- sage.arith.misc.is_pseudoprime(n)#
Test whether
n
is a pseudo-primeThe result is NOT proven correct - this is a pseudo-primality test!.
INPUT:
n
– an integer
Note
We do not consider negatives of prime numbers as prime.
EXAMPLES:
sage: is_pseudoprime(389) True sage: is_pseudoprime(2000) False sage: is_pseudoprime(2) True sage: is_pseudoprime(-1) False sage: factor(-6) -1 * 2 * 3 sage: is_pseudoprime(1) False sage: is_pseudoprime(-2) False
- sage.arith.misc.is_pseudoprime_power(n, get_data=False)#
Test if
n
is a power of a pseudoprime.The result is NOT proven correct - this IS a pseudo-primality test!. Note that a prime power is a positive power of a prime number so that 1 is not a prime power.
INPUT:
n
- an integerget_data
- (boolean) instead of a boolean return a pair \((p,k)\) so thatn
equals \(p^k\) and \(p\) is a pseudoprime or \((n,0)\) otherwise.
EXAMPLES:
sage: is_pseudoprime_power(389) True sage: is_pseudoprime_power(2000) False sage: is_pseudoprime_power(2) True sage: is_pseudoprime_power(1024) True sage: is_pseudoprime_power(-1) False sage: is_pseudoprime_power(1) False sage: is_pseudoprime_power(997^100) True
Use of the get_data keyword:
sage: is_pseudoprime_power(3^1024, get_data=True) (3, 1024) sage: is_pseudoprime_power(2^256, get_data=True) (2, 256) sage: is_pseudoprime_power(31, get_data=True) (31, 1) sage: is_pseudoprime_power(15, get_data=True) (15, 0)
Tests with numpy and gmpy2 numbers:
sage: from numpy import int16 sage: is_pseudoprime_power(int16(1024)) True sage: from gmpy2 import mpz sage: is_pseudoprime_power(mpz(1024)) True
- sage.arith.misc.is_square(n, root=False)#
Return whether or not
n
is square.If
n
is a square also return the square root. Ifn
is not square, also returnNone
.INPUT:
n
– an integerroot
– whether or not to also return a square root (default:False
)
OUTPUT:
bool
– whether or not a squareobject
– (optional) an actual square if found, andNone
otherwise.
EXAMPLES:
sage: is_square(2) False sage: is_square(4) True sage: is_square(2.2) True sage: is_square(-2.2) False sage: is_square(CDF(-2.2)) True sage: is_square((x-1)^2) Traceback (most recent call last): ... NotImplementedError: is_square() not implemented for non-constant or relational elements of Symbolic Ring
sage: is_square(4, True) (True, 2)
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: is_square(int8(4)) True sage: from gmpy2 import mpz sage: is_square(mpz(4)) True
Tests with Polynomial:
sage: R.<v> = LaurentPolynomialRing(QQ, 'v') sage: H = IwahoriHeckeAlgebra('A3', v**2) sage: R.<a,b,c,d> = QQ[] sage: p = a*b + c*d*a*d*a + 5 sage: is_square(p**2) True
- sage.arith.misc.is_squarefree(n)#
Test whether
n
is square free.EXAMPLES:
sage: is_squarefree(100) False sage: is_squarefree(101) True sage: R = ZZ['x'] sage: x = R.gen() sage: is_squarefree((x^2+x+1) * (x-2)) True sage: is_squarefree((x-1)**2 * (x-3)) False sage: O = ZZ[sqrt(-1)] sage: I = O.gen(1) sage: is_squarefree(I+1) True sage: is_squarefree(O(2)) False sage: O(2).factor() (-I) * (I + 1)^2
This method fails on domains which are not Unique Factorization Domains:
sage: O = ZZ[sqrt(-5)] sage: a = O.gen(1) sage: is_squarefree(a - 3) Traceback (most recent call last): ... ArithmeticError: non-principal ideal in factorization
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: is_squarefree(int8(100)) False sage: is_squarefree(int8(101)) True sage: from gmpy2 import mpz sage: is_squarefree(mpz(100)) False sage: is_squarefree(mpz(101)) True
- sage.arith.misc.jacobi_symbol(a, b)#
The Jacobi symbol of integers a and b, where b is odd.
Note
The
kronecker_symbol()
command extends the Jacobi symbol to all integers b.If
\(b = p_1^{e_1} * ... * p_r^{e_r}\)
then
\((a|b) = (a|p_1)^{e_1} ... (a|p_r)^{e_r}\)
where \((a|p_j)\) are Legendre Symbols.
INPUT:
a
- an integerb
- an odd integer
EXAMPLES:
sage: jacobi_symbol(10,777) -1 sage: jacobi_symbol(10,5) 0 sage: jacobi_symbol(10,2) Traceback (most recent call last): ... ValueError: second input must be odd, 2 is not odd
Tests with numpy and gmpy2 numbers:
sage: from numpy import int16 sage: jacobi_symbol(int16(10),int16(777)) -1 sage: from gmpy2 import mpz sage: jacobi_symbol(mpz(10),mpz(777)) -1
- sage.arith.misc.kronecker(x, y)#
The Kronecker symbol \((x|y)\).
INPUT:
x
– integery
– integer
OUTPUT:
an integer
EXAMPLES:
sage: kronecker_symbol(13,21) -1 sage: kronecker_symbol(101,4) 1
This is also available as
kronecker()
:sage: kronecker(3,5) -1 sage: kronecker(3,15) 0 sage: kronecker(2,15) 1 sage: kronecker(-2,15) -1 sage: kronecker(2/3,5) 1
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: kronecker_symbol(int8(13),int8(21)) -1 sage: from gmpy2 import mpz sage: kronecker_symbol(mpz(13),mpz(21)) -1
- sage.arith.misc.kronecker_symbol(x, y)#
The Kronecker symbol \((x|y)\).
INPUT:
x
– integery
– integer
OUTPUT:
an integer
EXAMPLES:
sage: kronecker_symbol(13,21) -1 sage: kronecker_symbol(101,4) 1
This is also available as
kronecker()
:sage: kronecker(3,5) -1 sage: kronecker(3,15) 0 sage: kronecker(2,15) 1 sage: kronecker(-2,15) -1 sage: kronecker(2/3,5) 1
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: kronecker_symbol(int8(13),int8(21)) -1 sage: from gmpy2 import mpz sage: kronecker_symbol(mpz(13),mpz(21)) -1
- sage.arith.misc.legendre_symbol(x, p)#
The Legendre symbol \((x|p)\), for \(p\) prime.
Note
The
kronecker_symbol()
command extends the Legendre symbol to composite moduli and \(p=2\).INPUT:
x
- integerp
- an odd prime number
EXAMPLES:
sage: legendre_symbol(2,3) -1 sage: legendre_symbol(1,3) 1 sage: legendre_symbol(1,2) Traceback (most recent call last): ... ValueError: p must be odd sage: legendre_symbol(2,15) Traceback (most recent call last): ... ValueError: p must be a prime sage: kronecker_symbol(2,15) 1 sage: legendre_symbol(2/3,7) -1
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: legendre_symbol(int8(2),int8(3)) -1 sage: from gmpy2 import mpz sage: legendre_symbol(mpz(2),mpz(3)) -1
- sage.arith.misc.mqrr_rational_reconstruction(u, m, T)#
Maximal Quotient Rational Reconstruction.
For research purposes only - this is pure Python, so slow.
INPUT:
u, m, T
- integers such that \(m > u \ge 0\), \(T > 0\).
OUTPUT:
Either integers \(n,d\) such that \(d>0\), \(\mathop{\mathrm{gcd}}(n,d)=1\), \(n/d=u \bmod m\), and \(T \cdot d \cdot |n| < m\), or
None
.Reference: Monagan, Maximal Quotient Rational Reconstruction: An Almost Optimal Algorithm for Rational Reconstruction (page 11)
This algorithm is probabilistic.
EXAMPLES:
sage: mqrr_rational_reconstruction(21,3100,13) (21, 1)
Tests with numpy and gmpy2 numbers:
sage: from numpy import int16 sage: mqrr_rational_reconstruction(int16(21),int16(3100),int16(13)) (21, 1) sage: from gmpy2 import mpz sage: mqrr_rational_reconstruction(mpz(21),mpz(3100),mpz(13)) (21, 1)
- sage.arith.misc.multinomial(*ks)#
Return the multinomial coefficient.
INPUT:
either an arbitrary number of integer arguments \(k_1,\dots,k_n\)
or an iterable (e.g. a list) of integers \([k_1,\dots,k_n]\)
OUTPUT:
Return the integer:
\[\binom{k_1 + \cdots + k_n}{k_1, \cdots, k_n} =\frac{\left(\sum_{i=1}^n k_i\right)!}{\prod_{i=1}^n k_i!} = \prod_{i=1}^n \binom{\sum_{j=1}^i k_j}{k_i}\]EXAMPLES:
sage: multinomial(0, 0, 2, 1, 0, 0) 3 sage: multinomial([0, 0, 2, 1, 0, 0]) 3 sage: multinomial(3, 2) 10 sage: multinomial(2^30, 2, 1) 618970023101454657175683075 sage: multinomial([2^30, 2, 1]) 618970023101454657175683075 sage: multinomial(Composition([1, 3])) 4 sage: multinomial(Partition([4, 2])) 15
AUTHORS:
Gabriel Ebner
- sage.arith.misc.multinomial_coefficients(m, n)#
Return a dictionary containing pairs \(\{(k_1, k_2, ..., k_m) : C_{k, n}\}\) where \(C_{k, n}\) are multinomial coefficients such that \(n = k_1 + k_2 + ...+ k_m\).
INPUT:
m
- integern
- integer
OUTPUT: dict
EXAMPLES:
sage: sorted(multinomial_coefficients(2, 5).items()) [((0, 5), 1), ((1, 4), 5), ((2, 3), 10), ((3, 2), 10), ((4, 1), 5), ((5, 0), 1)]
Notice that these are the coefficients of \((x+y)^5\):
sage: R.<x,y> = QQ[] sage: (x+y)^5 x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + y^5
sage: sorted(multinomial_coefficients(3, 2).items()) [((0, 0, 2), 1), ((0, 1, 1), 2), ((0, 2, 0), 1), ((1, 0, 1), 2), ((1, 1, 0), 2), ((2, 0, 0), 1)]
ALGORITHM: The algorithm we implement for computing the multinomial coefficients is based on the following result:
\[\binom{n}{k_1, \cdots, k_m} = \frac{k_1+1}{n-k_1}\sum_{i=2}^m \binom{n}{k_1+1, \cdots, k_i-1, \cdots}\]e.g.:
sage: k = (2, 4, 1, 0, 2, 6, 0, 0, 3, 5, 7, 1) # random value sage: n = sum(k) sage: s = 0 sage: for i in range(1, len(k)): ....: ki = list(k) ....: ki[0] += 1 ....: ki[i] -= 1 ....: s += multinomial(n, *ki) sage: multinomial(n, *k) == (k[0] + 1) / (n - k[0]) * s True
- sage.arith.misc.next_prime(n, proof=None)#
The next prime greater than the integer n. If n is prime, then this function does not return n, but the next prime after n. If the optional argument proof is False, this function only returns a pseudo-prime, as defined by the PARI nextprime function. If it is None, uses the global default (see
sage.structure.proof.proof
)INPUT:
n
- integerproof
- bool or None (default: None)
EXAMPLES:
sage: next_prime(-100) 2 sage: next_prime(1) 2 sage: next_prime(2) 3 sage: next_prime(3) 5 sage: next_prime(4) 5
Notice that the next_prime(5) is not 5 but 7.
sage: next_prime(5) 7 sage: next_prime(2004) 2011
- sage.arith.misc.next_prime_power(n)#
Return the smallest prime power greater than
n
.Note that if
n
is a prime power, then this function does not returnn
, but the next prime power aftern
.This function just calls the method
Integer.next_prime_power()
of Integers.See also
EXAMPLES:
sage: next_prime_power(1) 2 sage: next_prime_power(2) 3 sage: next_prime_power(10) 11 sage: next_prime_power(7) 8 sage: next_prime_power(99) 101
The same results can be obtained with:
sage: 1.next_prime_power() 2 sage: 2.next_prime_power() 3 sage: 10.next_prime_power() 11
Note that \(2\) is the smallest prime power:
sage: next_prime_power(-10) 2 sage: next_prime_power(0) 2
- sage.arith.misc.next_probable_prime(n)#
Return the next probable prime after self, as determined by PARI.
INPUT:
n
- an integer
EXAMPLES:
sage: next_probable_prime(-100) 2 sage: next_probable_prime(19) 23 sage: next_probable_prime(int(999999999)) 1000000007 sage: next_probable_prime(2^768) 1552518092300708935148979488462502555256886017116696611139052038026050952686376886330878408828646477950487730697131073206171580044114814391444287275041181139204454976020849905550265285631598444825262999193716468750892846853816058039
- sage.arith.misc.nth_prime(n)#
Return the n-th prime number (1-indexed, so that 2 is the 1st prime.)
INPUT:
n
– a positive integer
OUTPUT:
the n-th prime number
EXAMPLES:
sage: nth_prime(3) 5 sage: nth_prime(10) 29 sage: nth_prime(10^7) 179424673
sage: nth_prime(0) Traceback (most recent call last): ... ValueError: nth prime meaningless for non-positive n (=0)
- sage.arith.misc.number_of_divisors(n)#
Return the number of divisors of the integer n.
INPUT:
n
- a nonzero integer
OUTPUT:
an integer, the number of divisors of n
EXAMPLES:
sage: number_of_divisors(100) 9 sage: number_of_divisors(-720) 30
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: number_of_divisors(int8(100)) 9 sage: from gmpy2 import mpz sage: number_of_divisors(mpz(100)) 9
- sage.arith.misc.odd_part(n)#
The odd part of the integer \(n\). This is \(n / 2^v\), where \(v = \mathrm{valuation}(n,2)\).
EXAMPLES:
sage: odd_part(5) 5 sage: odd_part(4) 1 sage: odd_part(factorial(31)) 122529844256906551386796875
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: odd_part(int8(5)) 5 sage: from gmpy2 import mpz sage: odd_part(mpz(5)) 5
- sage.arith.misc.power_mod(a, n, m)#
Return the \(n\)-th power of \(a\) modulo \(m\), where \(a\) and \(m\) are elements of a ring that implements the modulo operator
%
.ALGORITHM: square-and-multiply
EXAMPLES:
sage: power_mod(2,388,389) 1 sage: power_mod(2,390,391) 285 sage: power_mod(2,-1,7) 4 sage: power_mod(11,1,7) 4
This function works for fairly general rings:
sage: R.<x> = ZZ[] sage: power_mod(3*x, 10, 7) 4*x^10 sage: power_mod(-3*x^2+4, 7, 2*x^3-5) x^14 + x^8 + x^6 + x^3 + 962509*x^2 - 791910*x - 698281
- sage.arith.misc.previous_prime(n)#
The largest prime < n. The result is provably correct. If n <= 1, this function raises a ValueError.
EXAMPLES:
sage: previous_prime(10) 7 sage: previous_prime(7) 5 sage: previous_prime(8) 7 sage: previous_prime(7) 5 sage: previous_prime(5) 3 sage: previous_prime(3) 2 sage: previous_prime(2) Traceback (most recent call last): ... ValueError: no previous prime sage: previous_prime(1) Traceback (most recent call last): ... ValueError: no previous prime sage: previous_prime(-20) Traceback (most recent call last): ... ValueError: no previous prime
- sage.arith.misc.previous_prime_power(n)#
Return the largest prime power smaller than
n
.The result is provably correct. If
n
is smaller or equal than2
this function raises an error.This function simply call the method
Integer.previous_prime_power()
of Integers.See also
EXAMPLES:
sage: previous_prime_power(3) 2 sage: previous_prime_power(10) 9 sage: previous_prime_power(7) 5 sage: previous_prime_power(127) 125
The same results can be obtained with:
sage: 3.previous_prime_power() 2 sage: 10.previous_prime_power() 9 sage: 7.previous_prime_power() 5 sage: 127.previous_prime_power() 125
Input less than or equal to \(2\) raises errors:
sage: previous_prime_power(2) Traceback (most recent call last): ... ValueError: no prime power less than 2 sage: previous_prime_power(-10) Traceback (most recent call last): ... ValueError: no prime power less than 2
sage: n = previous_prime_power(2^16 - 1) sage: while is_prime(n): ....: n = previous_prime_power(n) sage: factor(n) 251^2
- sage.arith.misc.prime_divisors(n)#
Return the list of prime divisors (up to units) of this element of a unique factorization domain.
INPUT:
n
– any object which can be decomposed into prime factors
OUTPUT:
A list of prime factors of
n
. For integers, this list is sorted in increasing order.EXAMPLES:
Prime divisors of positive integers:
sage: prime_divisors(1) [] sage: prime_divisors(100) [2, 5] sage: prime_divisors(2004) [2, 3, 167]
If
n
is negative, we do not include -1 among the prime divisors, since -1 is not a prime number:sage: prime_divisors(-100) [2, 5]
For polynomials we get all irreducible factors:
sage: R.<x> = PolynomialRing(QQ) sage: prime_divisors(x^12 - 1) [x - 1, x + 1, x^2 - x + 1, x^2 + 1, x^2 + x + 1, x^4 - x^2 + 1]
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: prime_divisors(int8(-100)) [2, 5] sage: from gmpy2 import mpz sage: prime_divisors(mpz(-100)) [2, 5]
- sage.arith.misc.prime_factors(n)#
Return the list of prime divisors (up to units) of this element of a unique factorization domain.
INPUT:
n
– any object which can be decomposed into prime factors
OUTPUT:
A list of prime factors of
n
. For integers, this list is sorted in increasing order.EXAMPLES:
Prime divisors of positive integers:
sage: prime_divisors(1) [] sage: prime_divisors(100) [2, 5] sage: prime_divisors(2004) [2, 3, 167]
If
n
is negative, we do not include -1 among the prime divisors, since -1 is not a prime number:sage: prime_divisors(-100) [2, 5]
For polynomials we get all irreducible factors:
sage: R.<x> = PolynomialRing(QQ) sage: prime_divisors(x^12 - 1) [x - 1, x + 1, x^2 - x + 1, x^2 + 1, x^2 + x + 1, x^4 - x^2 + 1]
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: prime_divisors(int8(-100)) [2, 5] sage: from gmpy2 import mpz sage: prime_divisors(mpz(-100)) [2, 5]
- sage.arith.misc.prime_powers(start, stop=None)#
List of all positive primes powers between
start
andstop
-1, inclusive. If the second argument is omitted, returns the prime powers up to the first argument.INPUT:
start
- an integer. If two inputs are given, a lower bound for the returned set of prime powers. If this is the only input, then it is an upper bound.stop
- an integer (default:None
). An upper bound for the returned set of prime powers.
OUTPUT:
The set of all prime powers between
start
andstop
or, if only one argument is passed, the set of all prime powers between 1 andstart
. The number \(n\) is a prime power if \(n=p^k\), where \(p\) is a prime number and \(k\) is a positive integer. Thus, \(1\) is not a prime power.EXAMPLES:
sage: prime_powers(20) [2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19] sage: len(prime_powers(1000)) 193 sage: len(prime_range(1000)) 168 sage: a = [z for z in range(95,1234) if is_prime_power(z)] sage: b = prime_powers(95,1234) sage: len(b) 194 sage: len(a) 194 sage: a[:10] [97, 101, 103, 107, 109, 113, 121, 125, 127, 128] sage: b[:10] [97, 101, 103, 107, 109, 113, 121, 125, 127, 128] sage: a == b True sage: prime_powers(100) == [i for i in range(100) if is_prime_power(i)] True sage: prime_powers(10,7) [] sage: prime_powers(-5) [] sage: prime_powers(-1,3) [2]
- sage.arith.misc.prime_to_m_part(n, m)#
Return the prime-to-
m
part ofn
.This is the largest divisor of
n
that is coprime tom
.INPUT:
n
– Integer (nonzero)m
– Integer
OUTPUT: Integer
EXAMPLES:
sage: prime_to_m_part(240,2) 15 sage: prime_to_m_part(240,3) 80 sage: prime_to_m_part(240,5) 48 sage: prime_to_m_part(43434,20) 21717
Note that integers also have a method with the same name:
sage: 240.prime_to_m_part(2) 15
Tests with numpy and gmpy2 numbers:
sage: from numpy import int16 sage: prime_to_m_part(int16(240), int16(2)) 15 sage: from gmpy2 import mpz sage: prime_to_m_part(mpz(240), mpz(2)) 15
- sage.arith.misc.primes(start, stop=None, proof=None)#
Return an iterator over all primes between start and stop-1, inclusive. This is much slower than
prime_range
, but potentially uses less memory. As withnext_prime()
, the optional argument proof controls whether the numbers returned are guaranteed to be prime or not.This command is like the Python 3
range
command, except it only iterates over primes. In some cases it is better to use primes thanprime_range
, because primes does not build a list of all primes in the range in memory all at once. However, it is potentially much slower since it simply calls thenext_prime()
function repeatedly, andnext_prime()
is slow.INPUT:
start
- an integer - lower bound for the primesstop
- an integer (or infinity) optional argument - giving upper (open) bound for the primesproof
- bool or None (default: None) If True, the function yields only proven primes. If False, the function uses a pseudo-primality test, which is much faster for really big numbers but does not provide a proof of primality. If None, uses the global default (seesage.structure.proof.proof
)
OUTPUT:
an iterator over primes from start to stop-1, inclusive
EXAMPLES:
sage: for p in primes(5,10): ....: print(p) 5 7 sage: list(primes(13)) [2, 3, 5, 7, 11] sage: list(primes(10000000000, 10000000100)) [10000000019, 10000000033, 10000000061, 10000000069, 10000000097] sage: max(primes(10^100, 10^100+10^4, proof=False)) 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000009631 sage: next(p for p in primes(10^20, infinity) if is_prime(2*p+1)) 100000000000000001243
- sage.arith.misc.primes_first_n(n, leave_pari=False)#
Return the first \(n\) primes.
INPUT:
\(n\) - a nonnegative integer
OUTPUT:
a list of the first \(n\) prime numbers.
EXAMPLES:
sage: primes_first_n(10) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] sage: len(primes_first_n(1000)) 1000 sage: primes_first_n(0) []
- sage.arith.misc.primitive_root(n, check=True)#
Return a positive integer that generates the multiplicative group of integers modulo \(n\), if one exists; otherwise, raise a
ValueError
.A primitive root exists if \(n=4\) or \(n=p^k\) or \(n=2p^k\), where \(p\) is an odd prime and \(k\) is a nonnegative number.
INPUT:
n
– a non-zero integercheck
– bool (default: True); if False, then \(n\) is assumed to be a positive integer possessing a primitive root, and behavior is undefined otherwise.
OUTPUT:
A primitive root of \(n\). If \(n\) is prime, this is the smallest primitive root.
EXAMPLES:
sage: primitive_root(23) 5 sage: primitive_root(-46) 5 sage: primitive_root(25) 2 sage: print([primitive_root(p) for p in primes(100)]) [1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5] sage: primitive_root(8) Traceback (most recent call last): ... ValueError: no primitive root
Note
It takes extra work to check if \(n\) has a primitive root; to avoid this, use
check=False
, which may slightly speed things up (but could also result in undefined behavior). For example, the second call below is an order of magnitude faster than the first:sage: n = 10^50 + 151 # a prime sage: primitive_root(n) 11 sage: primitive_root(n, check=False) 11
- sage.arith.misc.quadratic_residues(n)#
Return a sorted list of all squares modulo the integer \(n\) in the range \(0\leq x < |n|\).
EXAMPLES:
sage: quadratic_residues(11) [0, 1, 3, 4, 5, 9] sage: quadratic_residues(1) [0] sage: quadratic_residues(2) [0, 1] sage: quadratic_residues(8) [0, 1, 4] sage: quadratic_residues(-10) [0, 1, 4, 5, 6, 9] sage: v = quadratic_residues(1000); len(v) 159
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: quadratic_residues(int8(11)) [0, 1, 3, 4, 5, 9] sage: from gmpy2 import mpz sage: quadratic_residues(mpz(11)) [0, 1, 3, 4, 5, 9]
- sage.arith.misc.radical(n, *args, **kwds)#
Return the product of the prime divisors of n.
This calls
n.radical(*args, **kwds)
.EXAMPLES:
sage: radical(2 * 3^2 * 5^5) 30 sage: radical(0) Traceback (most recent call last): ... ArithmeticError: Radical of 0 not defined. sage: K.<i> = QuadraticField(-1) sage: radical(K(2)) i + 1
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: radical(int8(50)) 10 sage: from gmpy2 import mpz sage: radical(mpz(50)) 10
- sage.arith.misc.random_prime(n, proof=None, lbound=2)#
Return a random prime \(p\) between
lbound
and \(n\).The returned prime \(p\) satisfies
lbound
\(\leq p \leq n\).The returned prime \(p\) is chosen uniformly at random from the set of prime numbers less than or equal to \(n\).
INPUT:
n
- an integer >= 2.proof
- bool or None (default: None) If False, the function uses a pseudo-primality test, which is much faster for really big numbers but does not provide a proof of primality. If None, uses the global default (seesage.structure.proof.proof
)lbound
- an integer >= 2, lower bound for the chosen primes
EXAMPLES:
sage: p = random_prime(100000) sage: p.is_prime() True sage: p <= 100000 True sage: random_prime(2) 2
Here we generate a random prime between 100 and 200:
sage: p = random_prime(200, lbound=100) sage: p.is_prime() True sage: 100 <= p <= 200 True
If all we care about is finding a pseudo prime, then we can pass in
proof=False
sage: p = random_prime(200, proof=False, lbound=100) sage: p.is_pseudoprime() True sage: 100 <= p <= 200 True
AUTHORS:
Jon Hanke (2006-08-08): with standard Stein cleanup
Jonathan Bober (2007-03-17)
- sage.arith.misc.rational_reconstruction(a, m, algorithm='fast')#
This function tries to compute \(x/y\), where \(x/y\) is a rational number in lowest terms such that the reduction of \(x/y\) modulo \(m\) is equal to \(a\) and the absolute values of \(x\) and \(y\) are both \(\le \sqrt{m/2}\). If such \(x/y\) exists, that pair is unique and this function returns it. If no such pair exists, this function raises ZeroDivisionError.
An efficient algorithm for computing rational reconstruction is very similar to the extended Euclidean algorithm. For more details, see Knuth, Vol 2, 3rd ed, pages 656-657.
INPUT:
a
– an integerm
– a modulusalgorithm
– (default: ‘fast’)'fast'
- a fast implementation using direct GMP library calls in Cython.
OUTPUT:
Numerator and denominator \(n\), \(d\) of the unique rational number \(r=n/d\), if it exists, with \(n\) and \(|d| \le \sqrt{N/2}\). Return \((0,0)\) if no such number exists.
The algorithm for rational reconstruction is described (with a complete nontrivial proof) on pages 656-657 of Knuth, Vol 2, 3rd ed. as the solution to exercise 51 on page 379. See in particular the conclusion paragraph right in the middle of page 657, which describes the algorithm thus:
This discussion proves that the problem can be solved efficiently by applying Algorithm 4.5.2X with \(u=m\) and \(v=a\), but with the following replacement for step X2: If \(v3 \le \sqrt{m/2}\), the algorithm terminates. The pair \((x,y)=(|v2|,v3*\mathrm{sign}(v2))\) is then the unique solution, provided that \(x\) and \(y\) are coprime and \(x \le \sqrt{m/2}\); otherwise there is no solution. (Alg 4.5.2X is the extended Euclidean algorithm.)
Knuth remarks that this algorithm is due to Wang, Kornerup, and Gregory from around 1983.
EXAMPLES:
sage: m = 100000 sage: (119*inverse_mod(53,m))%m 11323 sage: rational_reconstruction(11323,m) 119/53
sage: rational_reconstruction(400,1000) Traceback (most recent call last): ... ArithmeticError: rational reconstruction of 400 (mod 1000) does not exist
sage: rational_reconstruction(3, 292393) 3 sage: a = Integers(292393)(45/97); a 204977 sage: rational_reconstruction(a, 292393, algorithm='fast') 45/97 sage: rational_reconstruction(293048, 292393) Traceback (most recent call last): ... ArithmeticError: rational reconstruction of 655 (mod 292393) does not exist sage: rational_reconstruction(0, 0) Traceback (most recent call last): ... ZeroDivisionError: rational reconstruction with zero modulus sage: rational_reconstruction(0, 1, algorithm="foobar") Traceback (most recent call last): ... ValueError: unknown algorithm 'foobar'
Tests with numpy and gmpy2 numbers:
sage: from numpy import int32 sage: rational_reconstruction(int32(3), int32(292393)) 3 sage: from gmpy2 import mpz sage: rational_reconstruction(mpz(3), mpz(292393)) 3
- sage.arith.misc.rising_factorial(x, a)#
Return the rising factorial \((x)^a\).
The notation in the literature is a mess: often \((x)^a\), but there are many other notations: GKP: Concrete Mathematics uses \(x^{\overline{a}}\).
The rising factorial is also known as the Pochhammer symbol, see Maple and Mathematica.
Definition: for integer \(a \ge 0\) we have \(x(x+1) \cdots (x+a-1)\). In all other cases we use the GAMMA-function: \(\frac {\Gamma(x+a)} {\Gamma(x)}\).
INPUT:
x
– element of a ringa
– a non-negative integer orx and a
– any numbers
OUTPUT: the rising factorial
See also
EXAMPLES:
sage: rising_factorial(10,3) 1320 sage: rising_factorial(10,RR('3.0')) 1320.00000000000 sage: rising_factorial(10,RR('3.3')) 2826.38895824964 sage: a = rising_factorial(1+I, I); a gamma(2*I + 1)/gamma(I + 1) sage: CC(a) 0.266816390637832 + 0.122783354006372*I sage: a = rising_factorial(I, 4); a -10 sage: x = polygen(ZZ) sage: rising_factorial(x, 4) x^4 + 6*x^3 + 11*x^2 + 6*x
AUTHORS:
Jaap Spies (2006-03-05)
- sage.arith.misc.sort_complex_numbers_for_display(nums)#
Given a list of complex numbers (or a list of tuples, where the first element of each tuple is a complex number), we sort the list in a “pretty” order.
Real numbers (with a zero imaginary part) come before complex numbers, and are sorted. Complex numbers are sorted by their real part unless their real parts are quite close, in which case they are sorted by their imaginary part.
This is not a useful function mathematically (not least because there is no principled way to determine whether the real components should be treated as equal or not). It is called by various polynomial root-finders; its purpose is to make doctest printing more reproducible.
We deliberately choose a cumbersome name for this function to discourage use, since it is mathematically meaningless.
EXAMPLES:
sage: import sage.arith.misc sage: sort_c = sort_complex_numbers_for_display sage: nums = [CDF(i) for i in range(3)] sage: for i in range(3): ....: nums.append(CDF(i + RDF.random_element(-3e-11, 3e-11), ....: RDF.random_element())) ....: nums.append(CDF(i + RDF.random_element(-3e-11, 3e-11), ....: RDF.random_element())) sage: shuffle(nums) sage: nums = sort_c(nums) sage: for i in range(len(nums)): ....: if nums[i].imag(): ....: first_non_real = i ....: break ....: else: ....: first_non_real = len(nums) sage: assert first_non_real >= 3 sage: for i in range(first_non_real - 1): ....: assert nums[i].real() <= nums[i + 1].real() sage: def truncate(n): ....: if n.real() < 1e-10: ....: return 0 ....: else: ....: return n.real().n(digits=9) sage: for i in range(first_non_real, len(nums)-1): ....: assert truncate(nums[i]) <= truncate(nums[i + 1]) ....: if truncate(nums[i]) == truncate(nums[i + 1]): ....: assert nums[i].imag() <= nums[i+1].imag()
- sage.arith.misc.squarefree_divisors(x)#
Return an iterator over the squarefree divisors (up to units) of this ring element.
Depends on the output of the prime_divisors function.
Squarefree divisors of an integer are not necessarily yielded in increasing order.
INPUT:
x – an element of any ring for which the prime_divisors function works.
EXAMPLES:
Integers with few prime divisors:
sage: list(squarefree_divisors(7)) [1, 7] sage: list(squarefree_divisors(6)) [1, 2, 3, 6] sage: list(squarefree_divisors(12)) [1, 2, 3, 6]
Squarefree divisors are not yielded in increasing order:
sage: list(squarefree_divisors(30)) [1, 2, 3, 6, 5, 10, 15, 30]
- sage.arith.misc.subfactorial(n)#
Subfactorial or rencontres numbers, or derangements: number of permutations of \(n\) elements with no fixed points.
INPUT:
n
- non negative integer
OUTPUT:
integer
- function value
EXAMPLES:
sage: subfactorial(0) 1 sage: subfactorial(1) 0 sage: subfactorial(8) 14833
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: subfactorial(int8(8)) 14833 sage: from gmpy2 import mpz sage: subfactorial(mpz(8)) 14833
AUTHORS:
Jaap Spies (2007-01-23)
- sage.arith.misc.sum_of_k_squares(k, n)#
Write the integer \(n\) as a sum of \(k\) integer squares if possible; otherwise raise a
ValueError
.INPUT:
k
– a non-negative integern
– an integer
OUTPUT: a tuple \((x_1, ..., x_k)\) of non-negative integers such that their squares sum to \(n\).
EXAMPLES:
sage: sum_of_k_squares(2, 9634) (15, 97) sage: sum_of_k_squares(3, 9634) (0, 15, 97) sage: sum_of_k_squares(4, 9634) (1, 2, 5, 98) sage: sum_of_k_squares(5, 9634) (0, 1, 2, 5, 98) sage: sum_of_k_squares(6, 11^1111-1) (19215400822645944253860920437586326284, 37204645194585992174252915693267578306, 3473654819477394665857484221256136567800161086815834297092488779216863122, 5860191799617673633547572610351797996721850737768032876360978911074629287841061578270832330322236796556721252602860754789786937515870682024273948, 20457423294558182494001919812379023992538802203730791019728543439765347851316366537094696896669915675685581905102118246887673397020172285247862426612188418787649371716686651256443143210952163970564228423098202682066311189439731080552623884051737264415984619097656479060977602722566383385989, 311628095411678159849237738619458396497534696043580912225334269371611836910345930320700816649653412141574887113710604828156159177769285115652741014638785285820578943010943846225597311231847997461959204894255074229895666356909071243390280307709880906261008237873840245959883405303580405277298513108957483306488193844321589356441983980532251051786704380984788999660195252373574924026139168936921591652831237741973242604363696352878914129671292072201700073286987126265965322808664802662993006926302359371379531571194266134916767573373504566621665949840469229781956838744551367172353) sage: sum_of_k_squares(7, 0) (0, 0, 0, 0, 0, 0, 0) sage: sum_of_k_squares(30,999999) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 7, 44, 999) sage: sum_of_k_squares(1, 9) (3,) sage: sum_of_k_squares(1, 10) Traceback (most recent call last): ... ValueError: 10 is not a sum of 1 square sage: sum_of_k_squares(1, -10) Traceback (most recent call last): ... ValueError: -10 is not a sum of 1 square sage: sum_of_k_squares(0, 9) Traceback (most recent call last): ... ValueError: 9 is not a sum of 0 squares sage: sum_of_k_squares(0, 0) () sage: sum_of_k_squares(7, -1) Traceback (most recent call last): ... ValueError: -1 is not a sum of 7 squares sage: sum_of_k_squares(-1, 0) Traceback (most recent call last): ... ValueError: k = -1 must be non-negative
Tests with numpy and gmpy2 numbers:
sage: from numpy import int16 sage: sum_of_k_squares(int16(2), int16(9634)) (15, 97) sage: from gmpy2 import mpz sage: sum_of_k_squares(mpz(2), mpz(9634)) (15, 97)
- sage.arith.misc.three_squares(n)#
Write the integer \(n\) as a sum of three integer squares if possible; otherwise raise a
ValueError
.INPUT:
n
– an integer
OUTPUT: a tuple \((a,b,c)\) of non-negative integers such that \(n = a^2 + b^2 + c^2\) with \(a <= b <= c\).
EXAMPLES:
sage: three_squares(389) (1, 8, 18) sage: three_squares(946) (9, 9, 28) sage: three_squares(2986) (3, 24, 49) sage: three_squares(7^100) (0, 0, 1798465042647412146620280340569649349251249) sage: three_squares(11^111-1) (616274160655975340150706442680, 901582938385735143295060746161, 6270382387635744140394001363065311967964099981788593947233) sage: three_squares(7 * 2^41) (1048576, 2097152, 3145728) sage: three_squares(7 * 2^42) Traceback (most recent call last): ... ValueError: 30786325577728 is not a sum of 3 squares sage: three_squares(0) (0, 0, 0) sage: three_squares(-1) Traceback (most recent call last): ... ValueError: -1 is not a sum of 3 squares
ALGORITHM:
- sage.arith.misc.trial_division(n, bound=None)#
Return the smallest prime divisor <= bound of the positive integer n, or n if there is no such prime. If the optional argument bound is omitted, then bound <= n.
INPUT:
n
- a positive integerbound
- (optional) a positive integer
OUTPUT:
int
- a prime p=bound that divides n, or n if there is no such prime.
EXAMPLES:
sage: trial_division(15) 3 sage: trial_division(91) 7 sage: trial_division(11) 11 sage: trial_division(387833, 300) 387833 sage: # 300 is not big enough to split off a sage: # factor, but 400 is. sage: trial_division(387833, 400) 389
Tests with numpy and gmpy2 numbers:
sage: from numpy import int8 sage: trial_division(int8(91)) 7 sage: from gmpy2 import mpz sage: trial_division(mpz(91)) 7
- sage.arith.misc.two_squares(n)#
Write the integer \(n\) as a sum of two integer squares if possible; otherwise raise a
ValueError
.INPUT:
n
– an integer
OUTPUT: a tuple \((a,b)\) of non-negative integers such that \(n = a^2 + b^2\) with \(a <= b\).
EXAMPLES:
sage: two_squares(389) (10, 17) sage: two_squares(21) Traceback (most recent call last): ... ValueError: 21 is not a sum of 2 squares sage: two_squares(21^2) (0, 21) sage: a,b = two_squares(100000000000000000129); a,b (4418521500, 8970878873) sage: a^2 + b^2 100000000000000000129 sage: two_squares(2^222+1) (253801659504708621991421712450521, 2583712713213354898490304645018692) sage: two_squares(0) (0, 0) sage: two_squares(-1) Traceback (most recent call last): ... ValueError: -1 is not a sum of 2 squares
ALGORITHM:
- sage.arith.misc.valuation(m, *args, **kwds)#
Return the valuation of
m
.This function simply calls the m.valuation() method. See the documentation of m.valuation() for a more precise description.
Note that the use of this functions is discouraged as it is better to use m.valuation() directly.
Note
This is not always a valuation in the mathematical sense. For more information see: sage.rings.finite_rings.integer_mod.IntegerMod_int.valuation
EXAMPLES:
sage: valuation(512,2) 9 sage: valuation(1,2) 0 sage: valuation(5/9, 3) -2
Valuation of 0 is defined, but valuation with respect to 0 is not:
sage: valuation(0,7) +Infinity sage: valuation(3,0) Traceback (most recent call last): ... ValueError: You can only compute the valuation with respect to a integer larger than 1.
Here are some other examples:
sage: valuation(100,10) 2 sage: valuation(200,10) 2 sage: valuation(243,3) 5 sage: valuation(243*10007,3) 5 sage: valuation(243*10007,10007) 1 sage: y = QQ['y'].gen() sage: valuation(y^3, y) 3 sage: x = QQ[['x']].gen() sage: valuation((x^3-x^2)/(x-4)) 2 sage: valuation(4r,2r) 2 sage: valuation(1r,1r) Traceback (most recent call last): ... ValueError: You can only compute the valuation with respect to a integer larger than 1. sage: from numpy import int16 sage: valuation(int16(512), int16(2)) 9 sage: from gmpy2 import mpz sage: valuation(mpz(512), mpz(2)) 9
- sage.arith.misc.xgcd(a, b)#
Return a triple
(g,s,t)
such that \(g = s\cdot a+t\cdot b = \gcd(a,b)\).Note
One exception is if \(a\) and \(b\) are not in a principal ideal domain (see Wikipedia article Principal_ideal_domain), e.g., they are both polynomials over the integers. Then this function can’t in general return
(g,s,t)
as above, since they need not exist. Instead, over the integers, we first multiply \(g\) by a divisor of the resultant of \(a/g\) and \(b/g\), up to sign.INPUT:
a, b
- integers or more generally, element of a ring for which the xgcd make sense (e.g. a field or univariate polynomials).
OUTPUT:
g, s, t
- such that \(g = s\cdot a + t\cdot b\)
Note
There is no guarantee that the returned cofactors (s and t) are minimal.
EXAMPLES:
sage: xgcd(56, 44) (4, 4, -5) sage: 4*56 + (-5)*44 4 sage: g, a, b = xgcd(5/1, 7/1); g, a, b (1, 3, -2) sage: a*(5/1) + b*(7/1) == g True sage: x = polygen(QQ) sage: xgcd(x^3 - 1, x^2 - 1) (x - 1, 1, -x) sage: K.<g> = NumberField(x^2-3) sage: g.xgcd(g+2) (1, 1/3*g, 0) sage: R.<a,b> = K[] sage: S.<y> = R.fraction_field()[] sage: xgcd(y^2, a*y+b) (1, a^2/b^2, ((-a)/b^2)*y + 1/b) sage: xgcd((b+g)*y^2, (a+g)*y+b) (1, (a^2 + (2*g)*a + 3)/(b^3 + g*b^2), ((-a + (-g))/b^2)*y + 1/b)
Here is an example of a xgcd for two polynomials over the integers, where the linear combination is not the gcd but the gcd multiplied by the resultant:
sage: R.<x> = ZZ[] sage: gcd(2*x*(x-1), x^2) x sage: xgcd(2*x*(x-1), x^2) (2*x, -1, 2) sage: (2*(x-1)).resultant(x) 2
Tests with numpy and gmpy2 types:
sage: from numpy import int8 sage: xgcd(4,int8(8)) (4, 1, 0) sage: xgcd(int8(4),int8(8)) (4, 1, 0) sage: from gmpy2 import mpz sage: xgcd(mpz(4), mpz(8)) (4, 1, 0) sage: xgcd(4, mpz(8)) (4, 1, 0)
- sage.arith.misc.xkcd(n='')#
This function is similar to the xgcd function, but behaves in a completely different way.
See https://xkcd.com/json.html for more details.
INPUT:
n
– an integer (optional)
OUTPUT: a fragment of HTML
EXAMPLES:
sage: xkcd(353) # optional - internet <h1>Python</h1><img src="https://imgs.xkcd.com/comics/python.png" title="I wrote 20 short programs in Python yesterday. It was wonderful. Perl, I'm leaving you."><div>Source: <a href="http://xkcd.com/353" target="_blank">http://xkcd.com/353</a></div>
- sage.arith.misc.xlcm(m, n)#
Extended lcm function: given two positive integers \(m,n\), returns a triple \((l,m_1,n_1)\) such that \(l=\mathop{\mathrm{lcm}}(m,n)=m_1 \cdot n_1\) where \(m_1|m\), \(n_1|n\) and \(\gcd(m_1,n_1)=1\), all with no factorization.
Used to construct an element of order \(l\) from elements of orders \(m,n\) in any group: see sage/groups/generic.py for examples.
EXAMPLES:
sage: xlcm(120,36) (360, 40, 9)