常用数论函数

cjsh

[这个贴子最后由cjsh在 2011/07/15 10:20am 第 1 次编辑] 常用数论函数 http://pari.math.u-bordeaux.fr/pub/pari/manuals/2.5.0/libpari.pdf 文档最后有全部函数索引 全开源才是数学家： http://pari.math.u-bordeaux.fr/dochtml/rluserman.html PARA是其它数学专业软件抄代码最多的，比较早的，影响很大，很有法国自由女神精神 MAGMA的：像求整函数 Truncate(n1) =12345678901111 AbsoluteValue(n) : RngIntElt -> RngIntElt Abs(n) : RngIntElt -> RngIntElt Absolute value of the integer n. Ilog2(n) : RngIntElt -> RngIntElt The integral part of the logarithm to the base two of the positive integer n. Ilog(b, n) : RngIntElt, RngIntElt -> RngIntElt The integral part of the logarithm to the base b of the positive integer n i.e., the largest integer k such that bk ≤n. The integer b must be greater than or equal to two. Quotrem(m, n) : RngIntElt, RngIntElt -> RngIntElt, RngIntElt Returns both the quotient q and remainder r obtained upon dividing the integer m by the integer n, that is, m = q.n + r, where 0 ≤r < n if n>0 and n RngIntElt, RngIntElt The valuation of the integer x at the prime p. This is the largest integer v for which pv divides x. If x = 0 then v = ∞. The optional second return value is the integer u such that x = pv u. Iroot(a, n) : RngIntElt, RngIntElt -> RngIntElt Given a positive integer a, return the integer b= ⌊root n of a⌋, i.e. the integral part of the n-th root of a. To obtain the actual root (as a real number), a must e coerced into a real field and the function Root applied. Sign(n) : RngIntElt -> RngIntElt Returns -1, 0 or 1 depending upon whether the integer n is negative, zero or positive, respectively. Ceiling(n) : RngIntElt -> RngIntElt The ceiling of the integer n, that is, n itself. Floor(n) : RngIntElt -> RngIntElt The floor of the integer n, that is, n itself. Round(n) : RngIntElt -> RngIntElt This function rounds the integer n to itself. Truncate(n) : RngIntElt -> RngIntElt This function returns the integer truncation of the integer n, that is, n itself. SquarefreeFactorization(n) : RngIntElt -> RngIntElt, RngIntElt Given a non-negative integer n, return a squarefree integer x as well as a positive integer y, such that n=xy2. Isqrt(n) : RngIntElt -> RngIntElt Lcm(s, t) : RngIntEltFact, RngIntEltFact -> RngIntEltFact Gcd(s, t) : RngIntEltFact, RngIntEltFact -> RngIntEltFact SquarefreeFactorization(f) : RngIntEltFact -> RngIntEltFact, RngIntEltFact MoebiusMu(f) : RngIntEltFact -> RngIntElt Divisors(f) : RngIntEltFact -> SeqEnum PrimeDivisors(f) : RngIntEltFact -> SeqEnum NumberOfDivisors(f) : RngIntEltFact -> RngIntElt SumOfDivisors(f) : RngIntEltFact -> RngIntElt IsOne(s) : RngIntEltFact -> BoolElt IsOdd(s) : RngIntEltFact -> BoolElt IsEven(s) : RngIntEltFact -> BoolElt IsUnit(s) : RngIntEltFact -> BoolElt IsPrime(s) : RngIntEltFact -> BoolElt IsPrimePower(s) : RngIntEltFact -> BoolElt IsSquare(s) : RngIntEltFact -> BoolElt IsSquarefree(s) : RngIntEltFact -> BoolElt Modexp(n, k, m) : RngIntElt, RngIntElt, RngIntElt -> RngIntElt The modular power nk mod m, where n is an integer, k is an integer and m is an integer greater than one. If k is negative, n must have an inverse i modulo m, and the result is then i - k mod m. The result is always an integer r with 0≤r< m. n mod m : RngIntElt, RngIntElt -> RngIntElt Remainder upon dividing the integer n by the integer m. The result always has the same sign as m. An error results if m is zero. Modinv(n, m) : RngIntElt, RngIntElt -> RngIntElt InverseMod(n, m) : RngIntElt, RngIntElt -> RngIntElt Given an integer n and a positive integer m, such that n and m are coprime, return an inverse u of n modulo m, that is, return an integer 1≤u BoolElt, RngIntElt Given an integer n and an integer m ≥2, this function returns an integer b such that 0 ≤b < m and b2 = n mod m if such b exists; an error results if no such root exists. Modorder(n, m) : RngIntElt, RngIntElt -> RngIntElt For integers n and m, m > 1, the function returns the least integer k ≥1 such that nk = 1 mod m, or zero if gcd(n, m) != 1. IsPrimitive(n, m) : RngIntElt, RngIntElt -> BoolElt Returns true if n is a primitive root for m, false otherwise (0 < n < m). PrimitiveRoot(m) : RngIntElt -> RngIntElt Given an integer m > 1, this function returns an integer value defined as follows: If Z/mZ has a primitive root and the function is successful in finding it, the root a is returned. If Z/mZ has a primitive root but the algorithm does not succeed in finding it, or Z/mZ does not possess a primitive root, then zero is returned. The Solution of Modular Equations The functions described here can be used if an occasional modular operation is required; the results are integers again. For more extensive modular arithmetic it is preferable to convert to residue class ring arithmetic. See section Residue Class Rings for details. Solution(a, b, m) : RngIntElt, RngIntElt, RngIntElt -> RngIntElt, RngIntElt If a solution exists to the linear congruence ax = b mod m, then returns x0, k such that x = x0 + i * k represents the complete set of solutions, where i can be any integer. Otherwise, returns -1. ChineseRemainderTheorem(X, N) : [RngIntElt], [RngIntElt] -> RngIntElt CRT(X, N) : [RngIntElt], [RngIntElt] -> RngIntElt Apply the Chinese Remainder Theorem to the integer sequences X and N. The sequences must have the same length, k say. The function returns the unique integer x in the range 0 ≤x < LCM(N[1].... .N[k]) such that x = X mod N. The elements of N must all be positive integers greater than one. If there is no solution, then -1 is returned. Solution(A, B, N) : [RngIntElt], [RngIntElt],[RngIntElt] -> RngIntElt Return a solution x to the system of simultaneous linear congruences defined by the integer sequences A, B and N. Each of these sequences must have the same number of terms, k say. The elements of N must all be positive integers greater than one. The i-th congruence is A .x = B mod N. The solution x will satisfy 0 ≤x < LCM(N[1].... .N[k]). If no solution exists, -1 is returned. NormEquation(d, m) : RngIntElt, RngIntElt -> BoolElt, RngIntElt, RngIntElt NormEquation(d, m: parameters)) : RngIntElt, RngIntElt -> BoolElt, RngIntElt, RngIntElt Factorization: [] Default: [ ]Given a positive integer d and a non-negative integer m, return true and two non-negative integers x and y, such that x2 + y2d = m, if such a solution exists. If such a solution does not exists only the value false is returned. If the factorization of m is known, it may be supplied as the value of the parameter Factorization to speed up the computation. Truncate(n1)