270 lines
6.1 KiB
C
270 lines
6.1 KiB
C
/* mpz_perfect_power_p(arg) -- Return non-zero if ARG is a perfect power,
|
|
zero otherwise.
|
|
|
|
Copyright 1998, 1999, 2000, 2001, 2005 Free Software Foundation, Inc.
|
|
|
|
Copyright 2008 Jason Moxham
|
|
|
|
This file is part of the GNU MP Library.
|
|
|
|
The GNU MP Library is free software; you can redistribute it and/or modify
|
|
it under the terms of the GNU Lesser General Public License as published by
|
|
the Free Software Foundation; either version 2.1 of the License, or (at your
|
|
option) any later version.
|
|
|
|
The GNU MP Library is distributed in the hope that it will be useful, but
|
|
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
|
|
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
|
|
License for more details.
|
|
|
|
You should have received a copy of the GNU Lesser General Public License
|
|
along with the GNU MP Library; see the file COPYING.LIB. If not, write to
|
|
the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
|
|
MA 02110-1301, USA. */
|
|
|
|
/*
|
|
We are to determine if c is a perfect power, c = a ^ b.
|
|
Assume c is divisible by 2^n and that codd = c/2^n is odd.
|
|
Assume a is divisible by 2^m and that aodd = a/2^m is odd.
|
|
It is always true that m divides n.
|
|
|
|
* If n is prime, either 1) a is 2*aodd and b = n
|
|
or 2) a = c and b = 1.
|
|
So for n prime, we readily have a solution.
|
|
* If n is factorable into the non-trivial factors p1,p2,...
|
|
Since m divides n, m has a subset of n's factors and b = n / m.
|
|
*/
|
|
|
|
/* This is a naive approach to recognizing perfect powers.
|
|
Many things can be improved. In particular, we should use p-adic
|
|
arithmetic for computing possible roots. */
|
|
|
|
#include <stdio.h> /* for NULL */
|
|
#include "mpir.h"
|
|
#include "gmp-impl.h"
|
|
#include "longlong.h"
|
|
|
|
static mpir_ui gcd _PROTO ((mpir_ui a, mpir_ui b));
|
|
static int isprime _PROTO ((mpir_ui t));
|
|
|
|
static const unsigned short primes[] =
|
|
{ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
|
|
59, 61, 67, 71, 73, 79, 83, 89, 97,101,103,107,109,113,127,131,
|
|
137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,
|
|
227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,
|
|
313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,
|
|
419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,
|
|
509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,
|
|
617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,
|
|
727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,
|
|
829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,
|
|
947,953,967,971,977,983,991,997,0
|
|
};
|
|
#define SMALLEST_OMITTED_PRIME 1009
|
|
|
|
|
|
int
|
|
mpz_perfect_power_p (mpz_srcptr u)
|
|
{
|
|
unsigned long int prime;
|
|
unsigned long int n, n2;
|
|
int i;
|
|
unsigned long int rem;
|
|
mpz_t u2, q;
|
|
int exact;
|
|
mp_size_t uns;
|
|
mp_size_t usize = SIZ (u);
|
|
TMP_DECL;
|
|
|
|
if (usize == 0)
|
|
return 1; /* consider 0 a perfect power */
|
|
|
|
n2 = mpz_scan1 (u, 0);
|
|
|
|
if (n2 == 1)
|
|
return 0; /* 2 divides exactly once. */
|
|
|
|
if (n2 > 1 && POW2_P(n2) && usize < 0)
|
|
return 0; /* 2 has power of two multiplicity with negative U */
|
|
|
|
TMP_MARK;
|
|
|
|
uns = ABS (usize) - n2 / BITS_PER_MP_LIMB;
|
|
MPZ_TMP_INIT (q, uns);
|
|
MPZ_TMP_INIT (u2, uns);
|
|
|
|
mpz_tdiv_q_2exp (u2, u, n2);
|
|
|
|
if (isprime (n2))
|
|
goto n2prime;
|
|
|
|
for (i = 1; primes[i] != 0; i++)
|
|
{
|
|
prime = primes[i];
|
|
|
|
if (mpz_divisible_ui_p (u2, prime)) /* divisible by this prime? */
|
|
{
|
|
rem = mpz_tdiv_q_ui (q, u2, prime * prime);
|
|
if (rem != 0)
|
|
{
|
|
TMP_FREE;
|
|
return 0; /* prime divides exactly once, reject */
|
|
}
|
|
mpz_swap (q, u2);
|
|
for (n = 2;;)
|
|
{
|
|
rem = mpz_tdiv_q_ui (q, u2, prime);
|
|
if (rem != 0)
|
|
break;
|
|
mpz_swap (q, u2);
|
|
n++;
|
|
}
|
|
|
|
if ( POW2_P(n) && usize < 0)
|
|
{
|
|
TMP_FREE;
|
|
return 0; /* power of two multiplicity with negative U, reject */
|
|
}
|
|
|
|
n2 = gcd (n2, n);
|
|
if (n2 == 1)
|
|
{
|
|
TMP_FREE;
|
|
return 0; /* we have multiplicity 1 of some factor */
|
|
}
|
|
|
|
if (mpz_cmpabs_ui (u2, 1) == 0)
|
|
{
|
|
TMP_FREE;
|
|
if(usize<0 && POW2_P(n2))return 0;/* factoring completed; not consistent power */
|
|
return 1; /* factoring completed; consistent power */
|
|
}
|
|
|
|
/* As soon as n2 becomes a prime number, stop factoring.
|
|
Either we have u=x^n2 or u is not a perfect power. */
|
|
if (isprime (n2))
|
|
goto n2prime;
|
|
}
|
|
}
|
|
|
|
if (n2 == 0)
|
|
{
|
|
/* We found no factors above; have to check all values of n. */
|
|
unsigned long int nth;
|
|
for (nth = usize < 0 ? 3 : 2;; nth++)
|
|
{
|
|
if (! isprime (nth))
|
|
continue;
|
|
#if 0
|
|
exact = mpz_padic_root (q, u2, nth, PTH);
|
|
if (exact)
|
|
#endif
|
|
exact = mpz_root (q, u2, nth);
|
|
if (exact)
|
|
{
|
|
TMP_FREE;
|
|
return 1;
|
|
}
|
|
if (mpz_cmpabs_ui (q, SMALLEST_OMITTED_PRIME) < 0)
|
|
{
|
|
TMP_FREE;
|
|
return 0;
|
|
}
|
|
}
|
|
}
|
|
else
|
|
{
|
|
unsigned long int nth;
|
|
/* We found some factors above. We just need to consider values of n
|
|
that divides n2. */
|
|
for (nth = usize < 0 ? 3 : 2; nth <= n2; nth++)
|
|
{
|
|
if (! isprime (nth))
|
|
continue;
|
|
if (n2 % nth != 0)
|
|
continue;
|
|
#if 0
|
|
exact = mpz_padic_root (q, u2, nth, PTH);
|
|
if (exact)
|
|
#endif
|
|
exact = mpz_root (q, u2, nth);
|
|
if (exact)
|
|
{
|
|
TMP_FREE;
|
|
return 1;
|
|
}
|
|
if (mpz_cmpabs_ui (q, SMALLEST_OMITTED_PRIME) < 0)
|
|
{
|
|
TMP_FREE;
|
|
return 0;
|
|
}
|
|
}
|
|
|
|
TMP_FREE;
|
|
return 0;
|
|
}
|
|
|
|
n2prime:
|
|
if(n2==2 && usize<0){TMP_FREE;return 0;}
|
|
exact = mpz_root (NULL, u2, n2);
|
|
TMP_FREE;
|
|
return exact;
|
|
}
|
|
|
|
static mpir_ui
|
|
gcd (mpir_ui a, mpir_ui b)
|
|
{
|
|
int an2, bn2, n2;
|
|
|
|
if (a == 0)
|
|
return b;
|
|
if (b == 0)
|
|
return a;
|
|
|
|
count_trailing_zeros (an2, a);
|
|
a >>= an2;
|
|
|
|
count_trailing_zeros (bn2, b);
|
|
b >>= bn2;
|
|
|
|
n2 = MIN (an2, bn2);
|
|
|
|
while (a != b)
|
|
{
|
|
if (a > b)
|
|
{
|
|
a -= b;
|
|
do
|
|
a >>= 1;
|
|
while ((a & 1) == 0);
|
|
}
|
|
else /* b > a. */
|
|
{
|
|
b -= a;
|
|
do
|
|
b >>= 1;
|
|
while ((b & 1) == 0);
|
|
}
|
|
}
|
|
|
|
return a << n2;
|
|
}
|
|
|
|
static int
|
|
isprime (mpir_ui t)
|
|
{
|
|
mpir_ui q, r, d;
|
|
|
|
if (t < 3 || (t & 1) == 0)
|
|
return t == 2;
|
|
|
|
for (d = 3, r = 1; r != 0; d += 2)
|
|
{
|
|
q = t / d;
|
|
r = t - q * d;
|
|
if (q < d)
|
|
return 1;
|
|
}
|
|
return 0;
|
|
}
|