void
checkprimes (unsigned long p1, unsigned long p2, unsigned long p3)
{
mpz_t b, f;
if (p1 - 1 != p2 - 1 + p3 - 1)
{
printf ("checkprimes(%lu, %lu, %lu) wrong\n", p1, p2, p3);
printf (" %lu - 1 != %lu - 1 + %lu - 1 \n", p1, p2, p3);
abort ();
}
mpz_init (b);
mpz_init (f);
checkWilson (b, p1); /* b = (p1-1)! */
checkWilson (f, p2); /* f = (p2-1)! */
mpz_divexact (b, b, f);
checkWilson (f, p3); /* f = (p3-1)! */
mpz_divexact (b, b, f); /* b = (p1-1)!/((p2-1)!(p3-1)!) */
mpz_bin_uiui (f, p1 - 1, p2 - 1);
if (mpz_cmp (f, b) != 0)
{
printf ("checkprimes(%lu, %lu, %lu) wrong\n", p1, p2, p3);
printf (" got "); mpz_out_str (stdout, 10, b); printf("\n");
printf (" want "); mpz_out_str (stdout, 10, f); printf("\n");
abort ();
}
mpz_clear (b);
mpz_clear (f);
}
int bern41_acceptable(mpz_t n, UV r, UV s, mpz_t t1, mpz_t t2)
{
double scmp = ceil(sqrt( (r-1)/3.0 )) * mpz_log2(n);
UV d = (UV) (0.5 * (r-1));
UV i = (UV) (0.475 * (r-1));
UV j = i;
/* Ensure conditions are correct */
if (d > r-2) d = r-2;
if (i > d) i = d;
if (j > (r-2-d)) j = r-2-d;
mpz_bin_uiui(t2, 2*s, i);
mpz_bin_uiui(t1, d, i); mpz_mul(t2, t2, t1);
mpz_bin_uiui(t1, 2*s-i, j); mpz_mul(t2, t2, t1);
mpz_bin_uiui(t1, r-2-d, j); mpz_mul(t2, t2, t1);
return (mpz_log2(t2) >= scmp);
}
int main(int argc, char **argv)
{
unsigned long i, j;
int tally;
mpz_t n;
mpz_init(n);
tally = 0;
for (i = 1; i <= 100; i++)
for (j = 1; j <= i; j++) {
mpz_bin_uiui(n, i, j);
if (mpz_cmp_ui(n, 1000000) > 0)
tally++;
}
printf("%d\n", tally);
return EXIT_SUCCESS;
}
void
try_mpz_bin_uiui (mpz_srcptr want, unsigned long n, unsigned long k)
{
mpz_t got;
mpz_init (got);
mpz_bin_uiui (got, n, k);
if (mpz_cmp (got, want) != 0)
{
printf ("mpz_bin_uiui wrong\n");
printf (" n=%lu\n", n);
printf (" k=%lu\n", k);
printf (" got="); mpz_out_str (stdout, 10, got); printf ("\n");
printf (" want="); mpz_out_str (stdout, 10, want); printf ("\n");
abort();
}
mpz_clear (got);
}
int main(void)
{
/* The number of paths from the origin to (a, b) is given by the binomial
* coefficient (a + b over b), which in this case is (40 over 20).
* This is equal to 40! / 20! * (40 - 20)!
*/
mpz_t x;
mpz_init(x);
mpz_bin_uiui(x, 40, 20);
mpz_out_str(stdout, 10, x);
mpz_clear(x);
return 0;
}
static PyObject *
GMPy_MPZ_Function_Bincoef(PyObject *self, PyObject *args)
{
MPZ_Object *result = NULL, *tempx;
unsigned long n, k;
if (PyTuple_GET_SIZE(args) != 2) {
TYPE_ERROR("bincoef() requires two integer arguments");
return NULL;
}
if (!(result = GMPy_MPZ_New(NULL))) {
/* LCOV_EXCL_START */
return NULL;
/* LCOV_EXCL_STOP */
}
k = c_ulong_From_Integer(PyTuple_GET_ITEM(args, 1));
if (k == (unsigned long)(-1) && PyErr_Occurred()) {
Py_DECREF((PyObject*)result);
return NULL;
}
n = c_ulong_From_Integer(PyTuple_GET_ITEM(args, 0));
if (!(n == (unsigned long)(-1) && PyErr_Occurred())) {
/* Use mpz_bin_uiui which should be faster. */
mpz_bin_uiui(result->z, n, k);
return (PyObject*)result;
}
if (!(tempx = GMPy_MPZ_From_Integer(PyTuple_GET_ITEM(args, 0), NULL))) {
Py_DECREF((PyObject*)result);
return NULL;
}
mpz_bin_ui(result->z, tempx->z, k);
Py_DECREF((PyObject*)tempx);
return (PyObject*)result;
}
/* returns 0 on success */
int
gen_consts (int numb, int nail, int limb)
{
mpz_t x, mask, y, last;
unsigned long a, b;
unsigned long ofl, ofe;
printf ("/* This file is automatically generated by gen-fac.c */\n\n");
printf ("#if GMP_NUMB_BITS != %d\n", numb);
printf ("Error , error this data is for %d GMP_NUMB_BITS only\n", numb);
printf ("#endif\n");
#if 0
printf ("#if GMP_LIMB_BITS != %d\n", limb);
printf ("Error , error this data is for %d GMP_LIMB_BITS only\n", limb);
printf ("#endif\n");
#endif
printf
("/* This table is 0!,1!,2!,3!,...,n! where n! has <= GMP_NUMB_BITS bits */\n");
printf
("#define ONE_LIMB_FACTORIAL_TABLE CNST_LIMB(0x1),CNST_LIMB(0x1");
mpz_init_set_ui (x, 1);
mpz_init (last);
for (b = 2;; b++)
{
mpz_mul_ui (x, x, b); /* so b!=a */
if (mpz_sizeinbase (x, 2) > numb)
break;
printf ("),CNST_LIMB(0x");
mpz_out_str (stdout, 16, x);
}
printf (")\n");
printf
("\n/* This table is 0!,1!,2!/2,3!/2,...,n!/2^sn where n!/2^sn is an */\n");
printf
("/* odd integer for each n, and n!/2^sn has <= GMP_NUMB_BITS bits */\n");
printf
("#define ONE_LIMB_ODD_FACTORIAL_TABLE CNST_LIMB(0x1),CNST_LIMB(0x1),CNST_LIMB(0x1");
mpz_set_ui (x, 1);
for (b = 3;; b++)
{
for (a = b; (a & 1) == 0; a >>= 1);
mpz_swap (last, x);
mpz_mul_ui (x, last, a);
if (mpz_sizeinbase (x, 2) > numb)
break;
printf ("),CNST_LIMB(0x");
mpz_out_str (stdout, 16, x);
}
printf (")\n");
printf
("#define ODD_FACTORIAL_TABLE_MAX CNST_LIMB(0x");
mpz_out_str (stdout, 16, last);
printf (")\n");
ofl = b - 1;
printf
("#define ODD_FACTORIAL_TABLE_LIMIT (%lu)\n", ofl);
mpz_init2 (mask, numb + 1);
mpz_setbit (mask, numb);
mpz_sub_ui (mask, mask, 1);
printf
("\n/* Previous table, continued, values modulo 2^GMP_NUMB_BITS */\n");
printf
("#define ONE_LIMB_ODD_FACTORIAL_EXTTABLE CNST_LIMB(0x");
mpz_and (x, x, mask);
mpz_out_str (stdout, 16, x);
mpz_init (y);
mpz_bin_uiui (y, b, b/2);
b++;
for (;; b++)
{
for (a = b; (a & 1) == 0; a >>= 1);
if (a == b) {
mpz_divexact_ui (y, y, a/2+1);
mpz_mul_ui (y, y, a);
} else
mpz_mul_2exp (y, y, 1);
if (mpz_sizeinbase (y, 2) > numb)
break;
mpz_mul_ui (x, x, a);
mpz_and (x, x, mask);
printf ("),CNST_LIMB(0x");
mpz_out_str (stdout, 16, x);
}
printf (")\n");
ofe = b - 1;
printf
("#define ODD_FACTORIAL_EXTTABLE_LIMIT (%lu)\n", ofe);
printf
("\n/* This table is 1!!,3!!,...,(2n+1)!! where (2n+1)!! has <= GMP_NUMB_BITS bits */\n");
printf
("#define ONE_LIMB_ODD_DOUBLEFACTORIAL_TABLE CNST_LIMB(0x1");
mpz_set_ui (x, 1);
for (b = 3;; b+=2)
{
mpz_swap (last, x);
mpz_mul_ui (x, last, b);
if (mpz_sizeinbase (x, 2) > numb)
break;
printf ("),CNST_LIMB(0x");
mpz_out_str (stdout, 16, x);
}
printf (")\n");
printf
("#define ODD_DOUBLEFACTORIAL_TABLE_MAX CNST_LIMB(0x");
mpz_out_str (stdout, 16, last);
printf (")\n");
printf
("#define ODD_DOUBLEFACTORIAL_TABLE_LIMIT (%lu)\n", b - 2);
printf
("\n/* This table x_1, x_2,... contains values s.t. x_n^n has <= GMP_NUMB_BITS bits */\n");
printf
("#define NTH_ROOT_NUMB_MASK_TABLE (GMP_NUMB_MASK");
for (b = 2;b <= 8; b++)
{
mpz_root (x, mask, b);
printf ("),CNST_LIMB(0x");
mpz_out_str (stdout, 16, x);
}
printf (")\n");
mpz_add_ui (mask, mask, 1);
printf
("\n/* This table contains inverses of odd factorials, modulo 2^GMP_NUMB_BITS */\n");
printf
("\n/* It begins with (2!/2)^-1=1 */\n");
printf
("#define ONE_LIMB_ODD_FACTORIAL_INVERSES_TABLE CNST_LIMB(0x1");
mpz_set_ui (x, 1);
for (b = 3;b <= ofe - 2; b++)
{
for (a = b; (a & 1) == 0; a >>= 1);
mpz_mul_ui (x, x, a);
mpz_invert (y, x, mask);
printf ("),CNST_LIMB(0x");
mpz_out_str (stdout, 16, y);
}
printf (")\n");
ofe = (ofe / 16 + 1) * 16;
printf
("\n/* This table contains 2n-popc(2n) for small n */\n");
printf
("\n/* It begins with 2-1=1 (n=1) */\n");
printf
("#define TABLE_2N_MINUS_POPC_2N 1");
for (b = 4; b <= ofe; b += 2)
{
mpz_set_ui (x, b);
printf (",%lu",b - mpz_popcount (x));
}
printf ("\n");
printf
("#define TABLE_LIMIT_2N_MINUS_POPC_2N %lu\n", ofe + 1);
ofl = (ofl + 1) / 2;
printf
("#define ODD_CENTRAL_BINOMIAL_OFFSET (%lu)\n", ofl);
printf
("\n/* This table contains binomial(2k,k)/2^t */\n");
printf
("\n/* It begins with ODD_CENTRAL_BINOMIAL_TABLE_MIN */\n");
printf
("#define ONE_LIMB_ODD_CENTRAL_BINOMIAL_TABLE ");
for (b = ofl;; b++)
{
mpz_bin_uiui (x, 2 * b, b);
mpz_remove_twos (x);
if (mpz_sizeinbase (x, 2) > numb)
break;
if (b != ofl)
printf ("),");
printf("CNST_LIMB(0x");
mpz_out_str (stdout, 16, x);
}
printf (")\n");
ofe = b - 1;
printf
("#define ODD_CENTRAL_BINOMIAL_TABLE_LIMIT (%lu)\n", ofe);
printf
("\n/* This table contains the inverses of elements in the previous table. */\n");
printf
("#define ONE_LIMB_ODD_CENTRAL_BINOMIAL_INVERSE_TABLE CNST_LIMB(0x");
for (b = ofl; b <= ofe; b++)
{
mpz_bin_uiui (x, 2 * b, b);
mpz_remove_twos (x);
mpz_invert (x, x, mask);
mpz_out_str (stdout, 16, x);
if (b != ofe)
printf ("),CNST_LIMB(0x");
}
printf (")\n");
printf
("\n/* This table contains the values t in the formula binomial(2k,k)/2^t */\n");
printf
("#define CENTRAL_BINOMIAL_2FAC_TABLE ");
for (b = ofl; b <= ofe; b++)
{
mpz_bin_uiui (x, 2 * b, b);
printf ("%d", mpz_remove_twos (x));
if (b != ofe)
printf (",");
}
printf ("\n");
return 0;
}
int is_aks_prime(mpz_t n)
{
mpz_t *px, *py;
int retval;
UV i, s, r, a;
UV starta = 1;
int _verbose = get_verbose_level();
if (mpz_cmp_ui(n, 4) < 0)
return (mpz_cmp_ui(n, 1) <= 0) ? 0 : 1;
/* Just for performance: check small divisors: 2*3*5*7*11*13*17*19*23 */
if (mpz_gcd_ui(0, n, 223092870UL) != 1 && mpz_cmp_ui(n, 23) > 0)
return 0;
if (mpz_perfect_power_p(n))
return 0;
#if AKS_VARIANT == AKS_VARIANT_V6 /* From the V6 AKS paper */
{
mpz_t sqrtn, t;
double log2n;
UV limit, startr;
PRIME_ITERATOR(iter);
mpz_init(sqrtn);
mpz_sqrt(sqrtn, n);
log2n = mpz_log2(n);
limit = (UV) floor( log2n * log2n );
if (_verbose>1) gmp_printf("# AKS checking order_r(%Zd) to %"UVuf"\n", n, (unsigned long) limit);
/* Using a native r limits us to ~2000 digits in the worst case (r ~ log^5n)
* but would typically work for 100,000+ digits (r ~ log^3n). This code is
* far too slow to matter either way. Composite r is ok here, but it will
* always end up prime, so save time and just check primes. */
retval = 0;
/* Start order search at a good spot. Idea from Nemana and Venkaiah. */
startr = (mpz_sizeinbase(n,2)-1) * (mpz_sizeinbase(n,2)-1);
startr = (startr < 1002) ? 2 : startr - 100;
for (r = 2; /* */; r = prime_iterator_next(&iter)) {
if (mpz_divisible_ui_p(n, r) ) /* r divides n. composite. */
{ retval = 0; break; }
if (mpz_cmp_ui(sqrtn, r) <= 0) /* no r <= sqrtn divides n. prime. */
{ retval = 1; break; }
if (r < startr) continue;
if (mpz_order_ui(r, n, limit) > limit)
{ retval = 2; break; }
}
prime_iterator_destroy(&iter);
mpz_clear(sqrtn);
if (retval != 2) return retval;
/* Since r is prime, phi(r) = r-1. */
s = (UV) floor( sqrt(r-1) * log2n );
}
#elif AKS_VARIANT == AKS_VARIANT_BORNEMANN /* Bernstein + Voloch */
{
UV slim;
double c2, x;
/* small t = few iters of big poly. big t = many iters of small poly */
double const t = (mpz_sizeinbase(n, 2) <= 64) ? 32 : 40;
double const t1 = (1.0/((t+1)*log(t+1)-t*log(t)));
double const dlogn = mpz_logn(n);
mpz_t tmp;
PRIME_ITERATOR(iter);
mpz_init(tmp);
prime_iterator_setprime(&iter, (UV) (t1*t1 * dlogn*dlogn) );
r = prime_iterator_next(&iter);
while (!is_primitive_root_uiprime(n,r))
r = prime_iterator_next(&iter);
prime_iterator_destroy(&iter);
slim = (UV) (2*t*(r-1));
c2 = dlogn * floor(sqrt(r));
{ /* Binary search for first s in [1,slim] where x >= 0 */
UV bi = 1;
UV bj = slim;
while (bi < bj) {
s = bi + (bj-bi)/2;
mpz_bin_uiui(tmp, r+s-1, s);
x = mpz_logn(tmp) / c2 - 1.0;
if (x < 0) bi = s+1;
else bj = s;
}
s = bi-1;
}
s = (s+3) >> 1;
/* Bornemann checks factors up to (s-1)^2, we check to max(r,s) */
/* slim = (s-1)*(s-1); */
slim = (r > s) ? r : s;
if (_verbose > 1) printf("# aks trial to %"UVuf"\n", slim);
if (_GMP_trial_factor(n, 2, slim) > 1)
{ mpz_clear(tmp); return 0; }
mpz_sqrt(tmp, n);
if (mpz_cmp_ui(tmp, slim) <= 0)
{ mpz_clear(tmp); return 1; }
mpz_clear(tmp);
}
#elif AKS_VARIANT == AKS_VARIANT_BERN21
{ /* Bernstein 2003, theorem 2.1 (simplified) */
UV q;
double slim, scmp, x;
mpz_t t, t2;
PRIME_ITERATOR(iter);
mpz_init(t); mpz_init(t2);
r = s = 0;
while (1) {
/* todo: Check r|n and r >= sqrt(n) here instead of waiting */
if (mpz_cmp_ui(n, r) <= 0) break;
r = prime_iterator_next(&iter);
q = largest_factor(r-1);
mpz_set_ui(t, r);
mpz_powm_ui(t, n, (r-1)/q, t);
if (mpz_cmp_ui(t, 1) <= 0) continue;
scmp = 2 * floor(sqrt(r)) * mpz_log2(n);
slim = 20 * (r-1);
/* Check viability */
mpz_bin_uiui(t, q+slim-1, slim); if (mpz_log2(t) < scmp) continue;
for (s = 2; s < slim; s++) {
mpz_bin_uiui(t, q+s-1, s);
if (mpz_log2(t) > scmp) break;
}
if (s < slim) break;
}
mpz_clear(t); mpz_clear(t2);
prime_iterator_destroy(&iter);
if (_GMP_trial_factor(n, 2, s) > 1)
return 0;
}
#elif AKS_VARIANT == AKS_VARIANT_BERN22
{ /* Bernstein 2003, theorem 2.2 (simplified) */
UV q;
double slim, scmp, x;
mpz_t t, t2;
PRIME_ITERATOR(iter);
mpz_init(t); mpz_init(t2);
r = s = 0;
while (1) {
/* todo: Check r|n and r >= sqrt(n) here instead of waiting */
if (mpz_cmp_ui(n, r) <= 0) break;
r = prime_iterator_next(&iter);
if (!is_primitive_root_uiprime(n,r)) continue;
q = r-1; /* Since r is prime, phi(r) = r-1 */
scmp = 2 * floor(sqrt(r-1)) * mpz_log2(n);
slim = 20 * (r-1);
/* Check viability */
mpz_bin_uiui(t, q+slim-1, slim); if (mpz_log2(t) < scmp) continue;
for (s = 2; s < slim; s++) {
mpz_bin_uiui(t, q+s-1, s);
if (mpz_log2(t) > scmp) break;
}
if (s < slim) break;
}
mpz_clear(t); mpz_clear(t2);
prime_iterator_destroy(&iter);
if (_GMP_trial_factor(n, 2, s) > 1)
return 0;
}
#elif AKS_VARIANT == AKS_VARIANT_BERN23
{ /* Bernstein 2003, theorem 2.3 (simplified) */
UV q, d, limit;
double slim, scmp, sbin, x, log2n;
mpz_t t, t2;
PRIME_ITERATOR(iter);
mpz_init(t); mpz_init(t2);
log2n = mpz_log2(n);
limit = (UV) floor( log2n * log2n );
r = 2;
s = 0;
while (1) {
/* todo: Check r|n and r >= sqrt(n) here instead of waiting */
if (mpz_cmp_ui(n, r) <= 0) break;
r++;
UV gcd = mpz_gcd_ui(NULL, n, r);
if (gcd != 1) { mpz_clear(t); mpz_clear(t2); return 0; }
UV v = mpz_order_ui(r, n, limit);
if (v >= limit) continue;
mpz_set_ui(t2, r);
totient(t, t2);
q = mpz_get_ui(t);
UV phiv = q/v;
/* printf("phi(%lu)/v = %lu/%lu = %lu\n", r, q, v, phiv); */
/* This is extremely inefficient. */
/* Choose an s value we'd be happy with */
slim = 20 * (r-1);
/* Quick check to see if it could work with s=slim, d=1 */
mpz_bin_uiui(t, q+slim-1, slim);
sbin = mpz_log2(t);
if (sbin < 2*floor(sqrt(q))*log2n)
continue;
for (s = 2; s < slim; s++) {
mpz_bin_uiui(t, q+s-1, s);
sbin = mpz_log2(t);
if (sbin < 2*floor(sqrt(q))*log2n) continue; /* d=1 */
/* Check each number dividing phi(r)/v */
for (d = 2; d < phiv; d++) {
if ((phiv % d) != 0) continue;
scmp = 2 * d * floor(sqrt(q/d)) * log2n;
if (sbin < scmp) break;
}
/* if we did not exit early, this s worked for each d. This s wins. */
if (d >= phiv) break;
}
if (s < slim) break;
}
mpz_clear(t); mpz_clear(t2);
prime_iterator_destroy(&iter);
if (_GMP_trial_factor(n, 2, s) > 1)
return 0;
}
#elif AKS_VARIANT == AKS_VARIANT_BERN41
{
double const log2n = mpz_log2(n);
/* Tuning: Initial 'r' selection */
double const r0 = 0.008 * log2n * log2n;
/* Tuning: Try a larger 'r' if 's' looks very large */
UV const rmult = 8;
UV slim;
mpz_t tmp, tmp2;
PRIME_ITERATOR(iter);
mpz_init(tmp); mpz_init(tmp2);
/* r has to be at least 3. */
prime_iterator_setprime(&iter, (r0 < 2) ? 2 : (UV) r0);
r = prime_iterator_next(&iter);
/* r must be a primitive root. For performance, skip if s looks too big. */
while ( !is_primitive_root_uiprime(n, r) ||
!bern41_acceptable(n, r, rmult*(r-1), tmp, tmp2) )
r = prime_iterator_next(&iter);
prime_iterator_destroy(&iter);
{ /* Binary search for first s in [1,lim] where conditions met */
UV bi = 1;
UV bj = rmult * (r-1);
while (bi < bj) {
s = bi + (bj-bi)/2;
if (!bern41_acceptable(n,r,s,tmp,tmp2)) bi = s+1;
else bj = s;
}
s = bj;
/* Our S goes from 2 to s+1. */
starta = 2;
s = s+1;
}
/* printf("chose r=%lu s=%lu d = %lu i = %lu j = %lu\n", r, s, d, i, j); */
/* Check divisibility to s(s-1) to cover both gcd conditions */
slim = s * (s-1);
if (_verbose > 1) printf("# aks trial to %"UVuf"\n", slim);
if (_GMP_trial_factor(n, 2, slim) > 1)
{ mpz_clear(tmp); mpz_clear(tmp2); return 0; }
/* If we checked divisibility to sqrt(n), then it is prime. */
mpz_sqrt(tmp, n);
if (mpz_cmp_ui(tmp, slim) <= 0)
{ mpz_clear(tmp); mpz_clear(tmp2); return 1; }
/* Check b^(n-1) = 1 mod n for b in [2..s] */
if (_verbose > 1) printf("# aks checking fermat to %"UVuf"\n", s);
mpz_sub_ui(tmp2, n, 1);
for (i = 2; i <= s; i++) {
mpz_set_ui(tmp, i);
mpz_powm(tmp, tmp, tmp2, n);
if (mpz_cmp_ui(tmp, 1) != 0)
{ mpz_clear(tmp); mpz_clear(tmp2); return 0; }
}
mpz_clear(tmp); mpz_clear(tmp2);
}
#endif
if (_verbose) gmp_printf("# AKS %Zd. r = %"UVuf" s = %"UVuf"\n", n, (unsigned long) r, (unsigned long) s);
/* Create the three polynomials we will use */
New(0, px, r, mpz_t);
New(0, py, r, mpz_t);
if ( !px || !py )
croak("allocation failure\n");
for (i = 0; i < r; i++) {
mpz_init(px[i]);
mpz_init(py[i]);
}
retval = 1;
for (a = starta; a <= s; a++) {
retval = test_anr(a, n, r, px, py);
if (!retval) break;
if (_verbose>1) { printf("."); fflush(stdout); }
}
if (_verbose>1) { printf("\n"); fflush(stdout); };
/* Free the polynomials */
for (i = 0; i < r; i++) {
mpz_clear(px[i]);
mpz_clear(py[i]);
}
Safefree(px);
Safefree(py);
return retval;
}
