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cyclo.cpp
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/*
* This source distribution is placed into the public domain by its author, Jonathan Crombie.
* You may use it for whatever purpose, without charge, and without having to notify anyone.
* I disclaim any responsibility for any errors or lack of fitness for any purpose.
*
* cyclo Version 1.3.5 Dec 19, 2023
*
* (Dec 19, 2023. Add a check early on to filter out non-LMs.)
* (Oct 13, 2022. Smarter ComputeLM. Eliminate mpz_pow() call from within for loops.)
* (Jun 30, 2016. Removed the pari dependency.)
* (Sep 22, 2014. Improved EulerTotient() -- skip computing all composite factors.)
* (Jul 16, 2013. Changes version 1.03 --> 1.02:
* Added some error messages and did some refactoring.)
* (Jul 1, 2013. Changes version 1.01 --> 1.02:
* some optimizations. Instead of trial factoring up to sqrt(n) for RemoveSquares(),
* ComputeAllFactors() etc., now only compute the prime factorization and then permute
* factors to compute the remaining composite factors.)
*
* (Dec 9, 2012. Changes version 1.00 --> 1.01:
* minor tweak -- modify pari_init() to not build a large number of prime differences.
* this speeds up initialization quite a bit.)
*
* To build try:
* g++ cyclo.cpp -lgmp -o cyclo
*
* (You will need to have g++ and the gmp library already installed)
* (For linux bash shell for Windows 10, try:
sudo apt-get install g++ libgmp3-dev )
*
* This source file is currently located at:
* http://myfactors.mooo.com/source/cyclo.cpp
*
*/
#include <stdio.h>
#include <stdlib.h>
#include <limits.h>
#include <values.h>
#include <gmp.h>
#include <string.h>
#include <ctype.h>
struct factor_info {
mpz_t the_factor;
long occurrences;
long digits;
char factor_status;
};
struct factor_infos {
long count;
struct factor_info* the_factors;
};
void ConvertPerfectPowers( mpz_t, mpz_t, mpz_t, mpz_t, mpz_t );
int IsLM( mpz_t, mpz_t, char );
int ComputeLM( mpz_t, mpz_t, mpz_t, char, mpz_t, mpz_t );
int ComputeLucasCD( mpz_t, mpz_t**, long*, mpz_t**, long* );
char quickprimecheck( mpz_t );
void TFDivideOut( mpz_t, long, char*, mpz_t, struct factor_infos* );
void Factorize( mpz_t, struct factor_infos* );
void PermuteFactors( struct factor_infos*, struct factor_infos*, long, mpz_t );
void ComputeAllFactors( mpz_t, struct factor_infos* );
long ComputeOccurrences( mpz_t, mpz_t );
int Mobius( mpz_t );
void EulerTotient( mpz_t, mpz_t );
void Init_Factor_Infos( struct factor_infos* );
void AddFactorInfo( struct factor_infos*, mpz_t, long, char, long );
void Cleanup_Factor_Infos( struct factor_infos* );
long NumberOfDigits( mpz_t );
void mpz_pow( mpz_t, mpz_t, mpz_t );
int factor_info_cmpfunc( const void*, const void* );
unsigned char wheel7[48] = { 2, 4, 2, 4, 6, 2, 6, 4, 2, 4,
6, 6, 2, 6, 4, 2, 6, 4, 6, 8,
4, 2, 4, 2, 4, 8, 6, 4, 6, 2,
4, 6, 2, 6, 6, 4, 2, 4, 6, 2,
6, 4, 2, 4, 2, 10, 2, 10 };
int main(int argc, char* argv[] )
{
if ( argc != 4 ) {
printf("\n");
printf("cyclo Version 1.3.5\n\n");
printf("Usage: \"cyclo base exponent +|-\"");
printf("\n\n");
return -1;
}
mpz_t base;
mpz_init( base );
mpz_set_str( base, argv[1], 10 );
if ( mpz_cmp_ui( base, 2 ) < 0 ) {
printf( "\n\nbase must be >= 2. Aborting.\n\n" );
return -1;
}
mpz_t exponent;
mpz_init( exponent );
mpz_set_str( exponent, argv[2], 10 );
char c0 = argv[3][0];
if ( c0 != '-' && c0 != '+' ) {
printf( "\n\nError: Third argument must be either \'-\' or \'+\'. Aborting.\n\n" );
mpz_clear( exponent );
mpz_clear( base );
exit( -1 );
}
{ // Do a quick test to weed out non-LMs early on
mpz_t gcd;
mpz_init( gcd );
mpz_gcd( gcd, base, exponent );
if ( mpz_cmp_ui( gcd, 1 ) == 0 ) {
gmp_printf("\n\n%Zd^%Zd %c 1 is not an LM number. Aborting.\n\n", base, exponent, c0 );
mpz_clear( gcd );
mpz_clear( exponent );
mpz_clear( base );
return 0;
}
mpz_clear( gcd );
}
mpz_t ConvertedBase;
mpz_init( ConvertedBase );
mpz_t ConvertedExponent;
mpz_init( ConvertedExponent );
mpz_t SqrFreeBase;
mpz_init( SqrFreeBase );
ConvertPerfectPowers( base, exponent, ConvertedBase, ConvertedExponent, SqrFreeBase );
if ( IsLM( SqrFreeBase, ConvertedExponent, c0 ) ) {
mpz_t L;
mpz_init( L );
mpz_t M;
mpz_init( M );
ComputeLM( ConvertedBase, SqrFreeBase, ConvertedExponent, c0, L, M );
gmp_printf( "L=%Zd\n", L );
gmp_printf( "M=%Zd\n", M );
mpz_clear( M );
mpz_clear( L );
}
else {
gmp_printf("\n\n%Zd^%Zd %c 1 is not an LM number. Aborting.\n\n", base, exponent, c0 );
}
mpz_clear( ConvertedBase );
mpz_clear( ConvertedExponent );
mpz_clear( SqrFreeBase );
mpz_clear( exponent );
mpz_clear( base );
return 0;
}
// Need to fix base, exponent if base is a perfect power
// Compute the square-free part of base as well, since it's really easy to do.
void ConvertPerfectPowers( mpz_t base, mpz_t exponent, mpz_t ConvertedBase, mpz_t ConvertedExponent, mpz_t SqrFreeBase ) {
struct factor_infos PrimeFactorization;
Init_Factor_Infos( &PrimeFactorization );
Factorize( base, &PrimeFactorization );
mpz_t gcd;
mpz_init_set_ui( gcd, PrimeFactorization.the_factors[0].occurrences );
long i;
for ( i = 1; i < PrimeFactorization.count && mpz_cmp_ui( gcd, 1 ) != 0; i++ )
mpz_gcd_ui( gcd, gcd, PrimeFactorization.the_factors[i].occurrences );
long power = mpz_get_d( gcd );
mpz_set_ui( ConvertedBase, 1 );
mpz_mul_ui( ConvertedExponent, exponent, power );
mpz_t tempz1;
mpz_init( tempz1 );
mpz_set_ui( SqrFreeBase, 1 );
for ( i = 0; i < PrimeFactorization.count; i++ ) {
long reducedpower = PrimeFactorization.the_factors[i].occurrences / power;
mpz_pow_ui( tempz1, PrimeFactorization.the_factors[i].the_factor, reducedpower );
mpz_mul( ConvertedBase, ConvertedBase, tempz1 );
if ( reducedpower % 2 )
mpz_mul( SqrFreeBase, SqrFreeBase, PrimeFactorization.the_factors[i].the_factor );
}
mpz_clear( tempz1 );
mpz_clear( gcd );
Cleanup_Factor_Infos( &PrimeFactorization );
}
// Returns 1 if base^exponent + c0 has an Aurifeuillian factorization else 0.
// Assumption: base is not a perfect power and is square-free
int IsLM( mpz_t base, mpz_t exponent, char c0 ) {
// Not handled values
if ( mpz_cmp_ui( base, 2 ) < 0 ||
mpz_cmp_ui( exponent, 2 ) < 0 ||
(c0 != '-' && c0 != '+') ) {
return 0;
}
{ // check that we have the correct c0. c0 should be -1 if base == 1 mod 4, else c0 should be 1
mpz_t r;
mpz_init( r );
mpz_mod_ui( r, base, 4 );
if ( ( (c0 == '+') && mpz_cmp_ui( r, 1 ) == 0) ||
( (c0 == '-') && mpz_cmp_ui( r, 1 ) != 0) ) {
mpz_clear( r );
return 0;
}
mpz_clear( r );
}
if ( !mpz_divisible_p( exponent, base ) ) // not divisible. L,M do not exist
return 0;
mpz_t h;
mpz_init( h );
mpz_divexact( h, exponent, base );
if ( mpz_even_p( h ) ) { // h is even. L,M do not exist
mpz_clear( h );
return 0;
}
mpz_clear( h );
return 1;
}
// Compute C and D polynomial coefficients. Code implemented based on
// Richard Brent's "On computing factors of cyclotomic polynomials" Algorithm L.
int ComputeLucasCD( mpz_t n, mpz_t** C, long* C_count, mpz_t** D, long* D_count ) {
mpz_t tempz1;
mpz_init( tempz1 );
mpz_t tempz2;
mpz_init( tempz2 );
mpz_t nmod4;
mpz_init( nmod4 );
mpz_mod_ui( nmod4, n, 4 );
mpz_t nmod8;
mpz_init( nmod8 );
mpz_mod_ui( nmod8, n, 8 );
mpz_t nprime;
mpz_init( nprime );
if ( mpz_cmp_ui( nmod4, 1 ) == 0 )
mpz_set( nprime, n );
else
mpz_mul_ui( nprime, n, 2 );
mpz_t s;
mpz_init( s );
if ( mpz_cmp_ui( nmod4, 3 ) == 0 )
mpz_set_si( s, -1 );
else
mpz_set_ui( s, 1 );
mpz_t sprime;
mpz_init( sprime );
if ( mpz_cmp_ui( nmod8, 5 ) == 0 )
mpz_set_si( sprime, -1 );
else
mpz_set_ui( sprime, 1 );
mpz_t d;
mpz_init( d );
EulerTotient(nprime, tempz1 );
mpz_divexact_ui( d, tempz1, 2 );
// Step 1
long ld = mpz_get_ui( d );
mpz_t* q = (mpz_t*) calloc( ld + 1, sizeof( mpz_t ) );
long k;
for ( k = 0; k <= ld; k++ )
mpz_init_set_ui( q[k], 0 );
mpz_t mpz_k;
mpz_init( mpz_k );
for ( k = 1; k <= ld; k++ ) {
mpz_set_ui( mpz_k, k );
if ( k % 2 )
mpz_set_si( q[k], mpz_jacobi( n, mpz_k ) );
else {
mpz_t gprime_k;
mpz_init( gprime_k );
mpz_gcd( gprime_k, mpz_k, nprime );
mpz_t tempz1;
mpz_init( tempz1 );
mpz_divexact( tempz1, nprime, gprime_k );
int mobius_value = Mobius( tempz1 );
mpz_t mpz_totient_value;
mpz_init( mpz_totient_value );
EulerTotient( gprime_k, mpz_totient_value );
int cosine_value = 0;
{
mpz_sub_ui( tempz1, n, 1 );
mpz_mul( tempz1, tempz1, mpz_k );
mpz_t mod8;
mpz_init( mod8 );
mpz_mod_ui( mod8, tempz1, 8 );
// we won't check for odd values since k is even and therefore mod8 is also even
if ( mpz_cmp_ui( mod8, 0 ) == 0 )
cosine_value = 1;
else if ( mpz_cmp_ui( mod8, 2 ) == 0 )
cosine_value = 0;
else if ( mpz_cmp_ui( mod8, 4 ) == 0 )
cosine_value = -1;
else // mod8 == 6
cosine_value = 0;
mpz_clear( mod8 );
}
mpz_mul_si( tempz1, mpz_totient_value, mobius_value );
mpz_mul_si( tempz1, tempz1, cosine_value );
mpz_set( q[k], tempz1 );
mpz_clear( mpz_totient_value );
mpz_clear( tempz1 );
mpz_clear( gprime_k );
}
}
mpz_t* gamma = (mpz_t*) calloc( ld + 1, sizeof( mpz_t ) );
for ( k = 0; k <= ld; k++ )
mpz_init_set_ui( gamma[k], 0 );
mpz_t* delta = (mpz_t*) calloc( ld, sizeof( mpz_t ) );
for ( k = 0; k < ld; k++ )
mpz_init_set_ui( delta[k], 0 );
// Step 2
mpz_set_ui( gamma[0], 1 );
mpz_set_ui( delta[0], 1 );
// Step 3
int d_is_odd = ld % 2;
long max_gamma = d_is_odd ? ( ld - 1 ) / 2 : ld / 2;
long max_delta = d_is_odd ? ( ld - 1 ) / 2 : ( ld - 2 ) / 2;
for ( k = 1; k <= max_gamma; k++ ) {
// compute gamma[k]
long j = 0;
for ( j = 0; j <= k - 1; j++ ) {
long q_index = 2 * k - 2 * j - 1;
mpz_mul( tempz1, n, q[q_index] );
mpz_mul( tempz1, tempz1, delta[j] );
mpz_add( gamma[k], gamma[k], tempz1 );
q_index++;
mpz_mul( tempz1, q[q_index], gamma[j] );
mpz_sub( gamma[k], gamma[k], tempz1 );
}
mpz_divexact_ui( gamma[k], gamma[k], 2 * k );
if ( k > max_delta )
continue;
// compute delta[k]
for ( j = 0; j <= k - 1; j++ ) {
long q_index = 2 * k + 1 - 2 * j;
mpz_mul( tempz1, q[q_index], gamma[j] );
mpz_add( delta[k], delta[k], tempz1 );
q_index--;
mpz_mul( tempz1, q[q_index], delta[j] );
mpz_sub( delta[k], delta[k], tempz1 );
}
mpz_add( delta[k], delta[k], gamma[k] );
mpz_divexact_ui( delta[k], delta[k], 2 * k + 1 );
}
// Step 4
for ( k = max_gamma + 1; k <= ld; k++ )
mpz_set( gamma[k], gamma[ld-k] );
// Step 5
for ( k = max_delta + 1; k < ld; k++ )
mpz_set( delta[k], delta[ld-1-k] );
// Pass back
*C = gamma;
*C_count = ld + 1;
*D = delta;
*D_count = ld;
// clean up
mpz_clear( mpz_k );
if ( q != NULL ) {
for ( k = 0; k <= ld; k++ )
mpz_clear( q[k] );
free( q );
q = NULL;
}
mpz_clear( d );
mpz_clear( sprime );
mpz_clear( s );
mpz_clear( nprime );
mpz_clear( nmod8 );
mpz_clear( nmod4 );
mpz_clear( tempz2 );
mpz_clear( tempz1 );
return 0;
}
// Computes the Aurifeuillian factors L,M for a^x +/- 1. Returns 0 on success.
// Assumption: base is not a perfect power
int ComputeLM( mpz_t base, mpz_t sqrfreebase, mpz_t exponent, char c0, mpz_t L, mpz_t M ) {
// Not handled values
if ( mpz_cmp_ui( base, 2 ) < 0 ||
mpz_cmp_ui( exponent, 2 ) < 0 ||
(c0 != '-' && c0 != '+') ) {
return -1;
}
{ // check that we have the correct c0. c0 should be -1 if sqrfreebase == 1 mod 4, else c0 should be 1
mpz_t r;
mpz_init( r );
mpz_mod_ui( r, sqrfreebase, 4 );
if ( (c0 == '+' && mpz_cmp_ui( r, 1 ) == 0) ||
(c0 == '-' && mpz_cmp_ui( r, 1 ) != 0) ) {
mpz_clear( r );
return -1;
}
mpz_clear( r );
}
if ( !mpz_divisible_p( exponent, sqrfreebase ) ) // not divisible. L,M do not exist
return -1;
unsigned long h = 0;
{
mpz_t temph;
mpz_init( temph );
mpz_divexact( temph, exponent, sqrfreebase );
h = mpz_get_ui( temph );
mpz_clear( temph );
if ( (h % 2) == 0 ) // h is even. L,M do not exist
return -1;
}
mpz_t* C = NULL;
long C_count = 0;
mpz_t* D = NULL;
long D_count = 0;
if ( ComputeLucasCD( sqrfreebase, &C, &C_count, &D, &D_count ) != 0 ) {
gmp_printf("Error: Could not compute Lucas C,D polynomial coefficients for base (%Zd). Aborting.\n", sqrfreebase );
return -1;
}
mpz_t tempz1;
mpz_init( tempz1 );
unsigned long k = (h + 1) / 2;
// Both Polynomial A & B are initially computed as follows:
//
// coef[0] * base^(h*0) + coef[1] * base^(h*1) + coef[2] * base^(h*2) + ...
// = coef[0] * (base^h)^0 + coef[1] * (base^h)^1 + coef[2] * (base^h)^2 + ...
// = coef[0] * 1 + coef[1] * base^h + coef[2] * base^h * base^h + ...
//
// B polynomial is then multiplied by additional terms
//
// Then,
// L = A - B
// M = A + B
// NOTE: Reading coefficients from beginning to end is exactly the same as reading from the end to the beginning.
// eg. for base 31, C = { 1,15,43,83,125,151,169,173,173,169,151,125,83,43,15,1 } and D = { 1,5,11,19,25,29,31,31,31,29,25,19,11,5,1 }
// Also, D_count = C_count - 1
// Compute Polynomial A with the Lucas 'C' coefficients
mpz_t A;
mpz_init( A );
long i;
{
mpz_t base_h; // base^h
mpz_init( base_h );
mpz_pow_ui( base_h, base, h );
mpz_t base_raised; // base_h^i
mpz_init( base_raised );
for ( i = 0; i < C_count; i++ ) { // each iteration computes one term and then adds to A
if ( i == 0 )
mpz_set_ui( base_raised, 1 );
else if ( i == 1 )
mpz_set( base_raised, base_h );
else
mpz_mul( base_raised, base_raised, base_h );
mpz_mul( tempz1, base_raised, C[i] );
mpz_add( A, A, tempz1 );
}
mpz_clear( base_raised );
mpz_clear( base_h );
}
// Compute Polynomial B with Lucas 'D' coefficients
mpz_t B;
mpz_init( B );
{
mpz_t base_h; // base^h
mpz_init( base_h );
mpz_pow_ui( base_h, base, h );
mpz_t base_raised; // base_h^i
mpz_init( base_raised );
for ( i = 0; i < D_count; i++ ) { // each iteration computes one term and then adds to B
if ( i == 0 )
mpz_set_ui( base_raised, 1 );
else if ( i == 1 )
mpz_set( base_raised, base_h );
else
mpz_mul( base_raised, base_raised, base_h );
mpz_mul( tempz1, base_raised, D[i] );
mpz_add( B, B, tempz1 );
}
mpz_clear( base_raised );
mpz_clear( base_h );
}
if ( mpz_cmp( base, sqrfreebase ) != 0 ) {
mpz_divexact( tempz1, base, sqrfreebase );
mpz_sqrt( tempz1, tempz1 );
mpz_pow_ui( tempz1, tempz1, h );
mpz_mul( B, B, tempz1 );
}
mpz_pow_ui( tempz1, sqrfreebase, k );
mpz_mul( B, B, tempz1 );
mpz_sub( L, A, B );
mpz_add( M, A, B );
mpz_clear( B );
mpz_clear( A );
mpz_clear( tempz1 );
for ( i = 0; i < C_count; i++ )
mpz_clear( C[i] );
free( C );
C = NULL;
for ( i = 0; i < D_count; i++ )
mpz_clear( D[i] );
free( D );
D = NULL;
return 0;
}
char quickprimecheck( mpz_t the_number ) {
char retval = 'C';
if ( mpz_cmp_ui( the_number, 1 ) == 0 ) {
retval = 'N';
return retval;
}
int factor_type = mpz_probab_prime_p( the_number, 30 );
if ( factor_type != 0 )
retval = 'P';
return retval;
}
// divide out denominator from running_N
void TFDivideOut( mpz_t running_N, long ldenominator, char* running_N_status, mpz_t square_root, struct factor_infos* Factor_Infos ) {
mpz_t denominator;
mpz_init_set_ui( denominator, ldenominator );
long occurrences = ComputeOccurrences( running_N, denominator );
long i = 0;
for ( i = occurrences; i > 0; i-- )
mpz_divexact_ui( running_N, running_N, ldenominator );
*running_N_status = quickprimecheck( running_N );
if ( *running_N_status == 'N' )
mpz_set_ui( square_root, 1 );
else
mpz_sqrt( square_root, running_N );
AddFactorInfo( Factor_Infos, denominator, occurrences, 'P', NumberOfDigits( denominator ) );
mpz_clear( denominator );
}
// Compute the prime factorization of a general number where square_root( the_number ) < MAXLONG - 255
void Factorize( mpz_t the_number, struct factor_infos* Factor_Infos ) {
if ( Factor_Infos == NULL )
return;
// Wheel Factorization. eg. http://programmingpraxis.com/2009/05/08/wheel-factorization/
mpz_t running_N;
mpz_init_set( running_N, the_number );
char running_N_status = quickprimecheck( running_N );
if ( running_N_status == 'P' ) {
AddFactorInfo( Factor_Infos, running_N, 1, 'P', NumberOfDigits( running_N ) );
mpz_clear( running_N );
return;
}
mpz_t square_root;
mpz_init( square_root );
mpz_sqrt( square_root, running_N );
if ( mpz_cmp_ui( square_root, MAXLONG - 255 ) > 0 ) {
printf( "Internal Error: Trying to factorize too big a number.\n" );
mpz_clear( square_root );
mpz_clear( running_N );
return;
}
if ( mpz_divisible_ui_p( running_N, 2 ) != 0 ) {
TFDivideOut( running_N, 2, &running_N_status, square_root, Factor_Infos );
if ( running_N_status == 'P' ) {
AddFactorInfo( Factor_Infos, running_N, 1, 'P', NumberOfDigits( running_N ) );
mpz_clear( square_root );
mpz_clear( running_N );
return;
}
}
if ( mpz_divisible_ui_p( running_N, 3 ) != 0 ) {
TFDivideOut( running_N, 3, &running_N_status, square_root, Factor_Infos );
if ( running_N_status == 'P' ) {
AddFactorInfo( Factor_Infos, running_N, 1, 'P', NumberOfDigits( running_N ) );
mpz_clear( square_root );
mpz_clear( running_N );
return;
}
}
if ( mpz_divisible_ui_p( running_N, 5 ) != 0 ) {
TFDivideOut( running_N, 5, &running_N_status, square_root, Factor_Infos );
if ( running_N_status == 'P' ) {
AddFactorInfo( Factor_Infos, running_N, 1, 'P', NumberOfDigits( running_N ) );
mpz_clear( square_root );
mpz_clear( running_N );
return;
}
}
if ( mpz_divisible_ui_p( running_N, 7 ) != 0 ) {
TFDivideOut( running_N, 7, &running_N_status, square_root, Factor_Infos );
if ( running_N_status == 'P' ) {
AddFactorInfo( Factor_Infos, running_N, 1, 'P', NumberOfDigits( running_N ) );
mpz_clear( square_root );
mpz_clear( running_N );
return;
}
}
unsigned long i = 0;
unsigned long denominator = 11;
for ( ;
running_N_status == 'C';
denominator += wheel7[i], i = ( i == 47 ? 0 : i + 1 ) ) {
if ( mpz_divisible_ui_p( running_N, denominator ) != 0 )
TFDivideOut( running_N, denominator, &running_N_status, square_root, Factor_Infos );
}
if ( running_N_status == 'P' )
AddFactorInfo( Factor_Infos, running_N, 1, 'P', NumberOfDigits( running_N ) );
mpz_clear( square_root );
mpz_clear( running_N );
}
void PermuteFactors( struct factor_infos* permutefactors, struct factor_infos* allfactors, long lefttumbler, mpz_t leftproduct, mpz_t the_number ) {
long tumbler = lefttumbler + 1;
mpz_t currfactor;
mpz_init( currfactor );
mpz_t currproduct;
mpz_init( currproduct );
long exponent;
for ( exponent = 0; exponent <= permutefactors->the_factors[tumbler].occurrences; exponent++ ) {
// compute primefactor^exponent
if ( exponent == 0 )
mpz_set_ui( currfactor, 1 );
else if ( exponent == 1 )
mpz_set( currfactor, permutefactors->the_factors[tumbler].the_factor );
else
mpz_mul( currfactor, currfactor, permutefactors->the_factors[tumbler].the_factor );
mpz_mul( currproduct, leftproduct, currfactor );
if ( tumbler < permutefactors->count - 1 )
PermuteFactors( permutefactors, allfactors, tumbler, currproduct, the_number );
else if ( mpz_cmp_ui( currproduct, 1 ) != 0 && mpz_cmp( currproduct, the_number ) != 0 ) {
char status = quickprimecheck( currproduct );
long occurrences = ComputeOccurrences( the_number, currproduct );
AddFactorInfo( allfactors, currproduct, occurrences, status, NumberOfDigits( currproduct ) );
}
}
mpz_clear( currproduct );
mpz_clear( currfactor );
}
// Compute all factors of the_number except for 1 and the number itself
void ComputeAllFactors( mpz_t the_number, struct factor_infos* Factor_Infos ) {
if ( Factor_Infos == NULL )
return;
if ( quickprimecheck( the_number ) != 'C' )
return;
struct factor_infos primes_only;
Init_Factor_Infos( &primes_only );
Factorize( the_number, &primes_only );
mpz_t leftproduct;
mpz_init_set_ui( leftproduct, 1 );
PermuteFactors( &primes_only, Factor_Infos, -1, leftproduct, the_number );
mpz_clear( leftproduct );
Cleanup_Factor_Infos( &primes_only );
qsort( Factor_Infos->the_factors, Factor_Infos->count, sizeof(struct factor_info), factor_info_cmpfunc );
}
// Compute the number of times denominator divides evenly into numerator
long ComputeOccurrences( mpz_t numerator, mpz_t denominator ) {
// not handled values
if ( mpz_cmp_ui( denominator, 1 ) <= 0 )
return -1;
long occurrences = 0;
mpz_t quotient;
mpz_init_set( quotient, numerator );
while ( mpz_divisible_p( quotient, denominator ) ) {
mpz_divexact( quotient, quotient, denominator );
occurrences++;
}
mpz_clear( quotient );
return occurrences;
}
// Compute the Mobius function
int Mobius( mpz_t n ) {
if ( mpz_cmp_ui( n, 1 ) == 0 )
return 1;
long total_occurrences = 0;
int has_duplicate_occurrences = 0;
if ( quickprimecheck( n ) == 'P' ) {
total_occurrences++;
}
else {
struct factor_infos Factor_Infos;
Init_Factor_Infos( &Factor_Infos );
ComputeAllFactors( n, &Factor_Infos );
long i = 0;
for ( ; i < Factor_Infos.count; i++ ) {
if ( Factor_Infos.the_factors[i].factor_status == 'P' ) {
total_occurrences += Factor_Infos.the_factors[i].occurrences;
if ( Factor_Infos.the_factors[i].occurrences > 1 )
has_duplicate_occurrences = 1;
}
}
Cleanup_Factor_Infos( &Factor_Infos );
}
if ( has_duplicate_occurrences )
return 0;
if ( total_occurrences % 2 )
return -1;
return 1;
}
// Compute Euler's Totient function
// tot(mn) = tot(m)*tot(n)*d/tot(d) where d = gcd(m,n)
// see: http://en.wikipedia.org/wiki/Euler's_totient_function
void EulerTotient( mpz_t the_number, mpz_t tot ) {
if ( mpz_cmp_si( the_number, 0 ) < 0 ) {
mpz_set_si( tot, -1 );
return;
}
mpz_set_ui( tot, 1 );
// by convention, tot(0) is 1
if ( mpz_cmp_si( the_number, 0 ) == 0 || mpz_cmp_ui( the_number, 1 ) == 0 )
return;
if ( quickprimecheck( the_number ) == 'P' ) {
mpz_sub_ui( tot, the_number, 1 );
return;
}
struct factor_infos primefactorization;
Init_Factor_Infos( &primefactorization );
Factorize( the_number, &primefactorization );
mpz_t tempZ;
mpz_init( tempZ );
long i,j;
// tot(n^k) = n^(k-1) * tot(n) see: http://mathworld.wolfram.com/TotientFunction.html (14)
// and if n is prime, then this is just: tot(p^k) = p^(k-1) * (p - 1)
for ( i = 0; i < primefactorization.count; i++ ) {
for ( j = 1; j < primefactorization.the_factors[i].occurrences; j++ )
mpz_mul( tot, tot, primefactorization.the_factors[i].the_factor );
mpz_sub_ui( tempZ, primefactorization.the_factors[i].the_fac
8A0A
tor, 1 );
mpz_mul( tot, tot, tempZ );
}
mpz_clear( tempZ );
Cleanup_Factor_Infos( &primefactorization );
}
// Initialize factor_infos
void Init_Factor_Infos( struct factor_infos* Factor_Infos ) {
if ( Factor_Infos == NULL )
return;
Factor_Infos->count = 0;
Factor_Infos->the_factors = NULL;
}
// Add a factor to factor_infos structure
void AddFactorInfo( struct factor_infos* Factor_Infos, mpz_t the_factor, long occurrences, char factor_status, long numdigits ) {
long index = 0;
// allocate memory
if ( Factor_Infos->count == 0 ) {
Factor_Infos->count = 1;
Factor_Infos->the_factors = (struct factor_info*) calloc( 1, sizeof(struct factor_info) );
}
else {
Factor_Infos->count++;
Factor_Infos->the_factors = (struct factor_info*) realloc( Factor_Infos->the_factors, sizeof(struct factor_info) * Factor_Infos->count );
index = Factor_Infos->count - 1;
memset( &Factor_Infos->the_factors[index], 0, sizeof(struct factor_info) );
}
mpz_init( Factor_Infos->the_factors[index].the_factor );
mpz_set( Factor_Infos->the_factors[index].the_factor, the_factor );
Factor_Infos->the_factors[index].digits = numdigits;
Factor_Infos->the_factors[index].occurrences = occurrences;
Factor_Infos->the_factors[index].factor_status = factor_status;
}
// Free the memory allocated
void Cleanup_Factor_Infos( struct factor_infos* Factor_Infos ) {
if ( Factor_Infos == NULL )
return;
long i;
for ( i = 0; i < Factor_Infos->count; i++ )
mpz_clear( Factor_Infos->the_factors[i].the_factor );
if ( Factor_Infos->the_factors != NULL ) {
free( Factor_Infos->the_factors );
Factor_Infos->the_factors = NULL;
}
Factor_Infos->count = 0;
}
// Return the number of digits (base 10) of "the_number". Positive numbers only.
long NumberOfDigits( mpz_t the_number ) {
if ( mpz_cmp_si( the_number, 0 ) < 0 )
return -1;
char* p = NULL;
gmp_asprintf( &p, "%Zd", the_number );
long len = strlen( p );
if ( p != NULL ) {
free( p );
p = NULL;
}
return len;
}
// Just call mpz_pow_ui() with some error checking first
void mpz_pow( mpz_t result, mpz_t b, mpz_t p ) {
if ( mpz_cmp_ui( p, ULONG_MAX ) > 0 ) {
printf( "Error: exponent too large in power function.\n" );
return;
}
mpz_pow_ui( result, b, mpz_get_ui( p ) );
}
int factor_info_cmpfunc( const void* p1, const void* p2 ) {
return mpz_cmp( (*((struct factor_info*)p1)).the_factor, (*((struct factor_info*)p2)).the_factor );
}