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ramanujan_primes.c
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ramanujan_primes.c
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#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "ptypes.h"
#define FUNC_log2floor 1
#include "util.h"
#define FUNC_is_prime_in_sieve 1
#include "prime_nth_count.h"
#include "sieve.h"
#include "ramanujan_primes.h"
/******************************************************************************/
/* RAMANUJAN PRIMES */
/******************************************************************************/
/* For Ramanujan prime estimates:
* - counts are done via inverse nth, so only one thing to tune.
* - For nth tables, upper values ok if too high, lower values ok if too low.
* - both upper & lower empirically tested to 175e9 (175 thousand million),
* with a return value of over 10^13.
*/
/* These are playing loose with Sondow/Nicholson/Noe 2011 theorem 4.
* The last value should be rigorously checked using actual R_n values. */
static const uint32_t small_ram_upper_idx[] = {
3970,3980,5218,5221,5224,5226,5262,5270,5272,5278,5281,5553,5556,7432,
7449,7453,8580,8584,8607,12589,12603,12620,12729,18119,18134,18174,18289,
18300,18401,18419,25799,27247,27267,28663,39635,40061,40366,45338,51320,
64439,65566,65829,84761,89055,104959,107852,146968,151755,186499,217258,
223956,270700,332195,347223,440804,508096,565039,768276,828377,1090285,
1277320,1568165,1896508,2375799,3300765,4162908,5124977,6522443,9298256,
11406250, 15873245, 21307556, 29899174, 40666215,
57180770, 81543888, 119596564, 177392936, 266391665,
411512446, 646331578, 1043239835, 1723380058, UVCONST(2919198776),
UVCONST(4294967295)
};
#define SMALL_NRAM_UPPER_MULT 2852
#define SMALL_NRAM_UPPER (sizeof(small_ram_upper_idx)/sizeof(small_ram_upper_idx[0]))
#if BITS_PER_WORD == 64
static const UV large_ram_upper_idx[] = {
UVCONST( 2256197513), UVCONST( 2556868249), UVCONST( 2919198776),
/* 11071, 11070, 11069, 11068, 11067, 11066, 11065, 11064, 11063 */
UVCONST( 3371836636), UVCONST( 3874119737), UVCONST( 4467380631),
UVCONST( 5163817509), UVCONST( 5950413657), UVCONST( 6901033442),
UVCONST( 8015893438), UVCONST( 9322299866), UVCONST( 10845166831),
/* 11062, 11061, 11060, 11059, 11058, 11057, 11056, 11055, 11054 */
UVCONST( 12727569836), UVCONST( 14852585181), UVCONST( 17463419944),
UVCONST( 20585027534), UVCONST( 24252210453), UVCONST( 28704897522),
UVCONST( 34003499133), UVCONST( 40436019651), UVCONST( 48229247660),
/* 11053, 11052, 11051, 11050, 11049, 11048, 11047, 11046, 11045 */
UVCONST( 57558675911), UVCONST( 69028965312), UVCONST( 83015434548),
UVCONST( 100138535684), UVCONST( 121051505524), UVCONST( 146783829698),
UVCONST( 178727808587), UVCONST( 218113299173), UVCONST( 267104085772),
/* 11044, 11043, 11042, 11041, 11040, 11039, 11038, 11037, 11036 */
UVCONST( 328057281739), UVCONST( 404608665617), UVCONST( 500552556306),
UVCONST( 621794385742), UVCONST( 774739900202), UVCONST( 969943548548),
UVCONST( 1218276754392), UVCONST( 1536655221634), UVCONST( 1946308957195),
/* 11035, 11034, 11033, 11032, 11031, 11030, 11029, 11028, 11027 */
UVCONST( 2475456777850), UVCONST( 3162491651655), UVCONST( 4058282334559),
UVCONST( 5233096936468), UVCONST( 6776539822896), UVCONST( 8821085181511),
UVCONST( 11539712635284), UVCONST( 15171808426849), UVCONST( 20056581407599),
/* 11026, 11025, 11024, 11023, 11022, 11021, 11020, 11019, 11018 */
UVCONST( 26656864542121), UVCONST( 35627338984775), UVCONST( 47899755943330),
UVCONST( 64773009691258), UVCONST( 88134778026475), UVCONST(120680838280663),
UVCONST(166331208358410), UVCONST(230783974844445), UVCONST(322443487572932),
/* 11017, 11016, 11015, 11014, 11013, 11012, 11011, 11010, 11009 */
UVCONST(453738479744216), UVCONST(643248344602940), UVCONST(918867804392140),
UVCONST(1322953724888193),UVCONST(1920282116080684),
1.47*UVCONST(1920282116080684), /* Estimates for larger */
2.3*UVCONST(1920282116080684),
3.4*UVCONST(1920282116080684),
5.1*UVCONST(1920282116080684),
7.9*UVCONST(1920282116080684),
12.2*UVCONST(1920282116080684),
};
#define LARGE_NRAM_UPPER_MULT 11075
#define LARGE_NRAM_UPPER (sizeof(large_ram_upper_idx)/sizeof(large_ram_upper_idx[0]))
#endif
UV nth_ramanujan_prime_upper(UV n) {
UV i, mult, res;
if (n <= 2) return (n==0) ? 0 : (n==1) ? 2 : 11;
res = nth_prime_upper(3*n);
if (n < UVCONST(2256197512) || BITS_PER_WORD < 64) {
/* While p_3n is a complete upper bound, Rp_n tends to p_2n, and
* SNN(2011) theorem 4 shows how we can find (m,c) values where m < 1,
* Rn < m*p_3n for all n > c. Here we use various quantized m values
* and the table gives us c values where it applies. */
if (n < 20) mult = 3580;
else if (n < 98) mult = 3340;
else if (n < 1580) mult = 3040;
else if (n < 3242) mult = 2885;
else {
for (i = 0; i < SMALL_NRAM_UPPER; i++)
if (small_ram_upper_idx[i] > n)
break;
mult = SMALL_NRAM_UPPER_MULT-i;
}
if (res > (UV_MAX/mult)) res = (UV) (((long double) mult / 4096.0L) * res);
else res = (res * mult) >> 12;
#if BITS_PER_WORD == 64
} else {
for (i = 0; i < LARGE_NRAM_UPPER; i++)
if (large_ram_upper_idx[i] > n)
break;
mult = (LARGE_NRAM_UPPER_MULT-i);
if (res > (UV_MAX/mult)) res = (UV) (((long double) mult / 16384.0L) * res);
else res = (res * mult) >> 14;
#endif
}
if (n > 43 && n < 10000) {
/* Calculate upper bound using Srinivasan and Arés 2017 */
/* TODO We should construct a tighter bound like this. */
double s = (2 * (double)n) * (1.0L + 1.0L/ramanujan_sa_gn(n));
UV ps = nth_prime_upper( (UV) s );
if (ps < res)
res = ps;
}
return res;
}
static const uint32_t small_ram_lower_idx[] = {
2785, 2800, 4275, 5935, 6107, 8797, 9556, 13314, 13641, 20457, 23745,
34432, 50564, 69194, 97434, 149399, 224590, 337116, 514260, 804041,
1317612, 2340461, 4332796, 8393680, 17227225, 38996663, 94437897,
253560792, 763315838, UVCONST(2663598260), UVCONST(4294967295)
};
#define SMALL_NRAM_LOWER_MULT 557
#define SMALL_NRAM_LOWER (sizeof(small_ram_lower_idx)/sizeof(small_ram_lower_idx[0]))
#if BITS_PER_WORD == 64
static const UV large_ram_lower_idx[] = {
UVCONST( 2267483962), UVCONST( 2663598260), UVCONST( 3152476871),
UVCONST( 3742932857), UVCONST( 4446913643), UVCONST( 5298293978),
UVCONST( 6318053149), UVCONST( 7608807497), UVCONST( 9140758346),
UVCONST( 11015956390), UVCONST( 13351265915), UVCONST( 16199147294),
/* 4213, 4212, 4211, 4210, 4209, 4208, 4207, 4206, 4205 */
UVCONST( 19739499402), UVCONST( 24137542585), UVCONST( 29629560254),
UVCONST( 36435870727), UVCONST( 45085624406), UVCONST( 55940244390),
UVCONST( 69713814138), UVCONST( 87221199999), UVCONST( 109606558728),
/* 4204, 4203, 4202, 4201, 4200, 4199, 4198, 4197, 4196 */
UVCONST( 138227790751), UVCONST( 175290761423), UVCONST( 223132516788),
UVCONST( 285315117360), UVCONST( 366761235749), UVCONST( 473606049986),
UVCONST( 614858505562), UVCONST( 802552362351), UVCONST( 1052957884730),
/* 4195, 4194, 4193, 4192, 4191, 4190, 4189, 4188, 4187 */
UVCONST( 1389550174208), UVCONST( 1843854433659), UVCONST( 2461728402552),
UVCONST( 3306766457564), UVCONST( 4469341663210), UVCONST( 6080948095909),
UVCONST( 8329279118918), UVCONST( 11488848759561), UVCONST( 15959135388235),
/* 4186, 4185, 4184, 4183, 4182, 4181, 4180, 4179, 4178 */
UVCONST( 22336622435614), UVCONST( 31501671598985), UVCONST( 44779902229212),
UVCONST( 64180867011184), UVCONST( 92772523880955), UVCONST(135282253437392),
UVCONST(199079826917291), UVCONST(295746797998912), UVCONST(443667118326600),
/* 4177, 4176, 4175, 4174, 4173, 4172, 4171, 4170, 4169 */
UVCONST(672350086039900),UVCONST(1029719394152693),UVCONST(1594365662292999),
1.55*UVCONST(1594365662292999), /* estimates here and further */
2.45*UVCONST(1594365662292999),
3.90*UVCONST(1594365662292999),
6.30*UVCONST(1594365662292999),
10.4*UVCONST(1594365662292999),
17.2*UVCONST(1594365662292999),
};
#define LARGE_NRAM_LOWER_MULT 4225
#define LARGE_NRAM_LOWER (sizeof(large_ram_lower_idx)/sizeof(large_ram_lower_idx[0]))
#endif
UV nth_ramanujan_prime_lower(UV n) {
UV res, i, mult;
if (n <= 2) return (n==0) ? 0 : (n==1) ? 2 : 11;
res = nth_prime_lower(2*n);
if (n < UVCONST(2267483962) || BITS_PER_WORD < 64) {
for (i = 0; i < SMALL_NRAM_LOWER; i++)
if (small_ram_lower_idx[i] > n)
break;
mult = (SMALL_NRAM_LOWER_MULT-i);
if (res > (UV_MAX/mult)) res = (UV) (((long double) mult / 512.0L) * res);
else res = (res * mult) >> 9;
#if BITS_PER_WORD == 64
} else {
if (n < large_ram_lower_idx[LARGE_NRAM_LOWER-1]) {
for (i = 0; i < LARGE_NRAM_LOWER; i++)
if (large_ram_lower_idx[i] > n)
break;
mult = (LARGE_NRAM_LOWER_MULT-i);
if (res > (UV_MAX/mult)) res = (UV) (((long double) mult / 4096.0L) * res);
else res = (res * mult) >> 12;
}
#endif
}
return res;
}
/* An advantage of making these binary searches on the inverse is that we
* don't have to tune them separately, and nothing changes if the prime
* count bounds are modified. We do need to keep up to date with any
* changes to nth_prime_{lower,upper} however. */
UV ramanujan_prime_count_lower(UV n) {
UV lo, hi;
if (n < 29) return (n < 2) ? 0 : (n < 11) ? 1 : (n < 17) ? 2 : 3;
/* Binary search on nth_ramanujan_prime_upper */
/* We know we're between p_2n and p_3n, probably close to the former. */
lo = prime_count_lower(n)/3;
hi = prime_count_upper(n) >> 1;
while (lo < hi) {
UV mid = lo + (hi-lo)/2;
if (nth_ramanujan_prime_upper(mid) < n) lo = mid+1;
else hi = mid;
}
return lo-1;
}
UV ramanujan_prime_count_upper(UV n) {
/* return prime_count_upper(n) >> 1; */ /* Simple bound */
UV lo, hi;
if (n < 29) return (n < 2) ? 0 : (n < 11) ? 1 : (n < 17) ? 2 : 3;
/* Binary search on nth_ramanujan_prime_upper */
/* We know we're between p_2n and p_3n, probably close to the former. */
lo = prime_count_lower(n)/3;
hi = prime_count_upper(n) >> 1;
while (lo < hi) {
UV mid = lo + (hi-lo)/2;
if (nth_ramanujan_prime_lower(mid) < n) lo = mid+1;
else hi = mid;
}
return lo-1;
}
/* Return array of first n ramanujan primes. Use Noe's algorithm. */
UV* n_ramanujan_primes(UV n) {
UV max, k, s, *L;
unsigned char* sieve;
max = nth_ramanujan_prime_upper(n); /* Rn <= max, so we can sieve to there */
if (_XS_get_verbose() >= 2) { printf("sieving to %"UVuf" for first %"UVuf" Ramanujan primes\n", max, n); fflush(stdout); }
Newz(0, L, n, UV);
L[0] = 2;
sieve = sieve_erat30(max);
for (s = 0, k = 7; k <= max; k += 2) {
if (is_prime_in_sieve(sieve, k)) s++;
if (s < n) L[s] = k+1;
if ((k & 3) == 1 && is_prime_in_sieve(sieve, (k+1)>>1)) s--;
if (s < n) L[s] = k+2;
}
Safefree(sieve);
return L;
}
UV* n_range_ramanujan_primes(UV nlo, UV nhi) {
UV mink, maxk, k, s, *L;
int verbose = _XS_get_verbose();
if (nlo == 0) nlo = 1;
if (nhi == 0) nhi = 1;
/* If we're starting from 1, just do single monolithic sieve */
if (nlo == 1) return n_ramanujan_primes(nhi);
Newz(0, L, nhi-nlo+1, UV);
if (nlo <= 1 && nhi >= 1) L[1-nlo] = 2;
if (nlo <= 2 && nhi >= 2) L[2-nlo] = 11;
if (nhi < 3) return L;
mink = nth_ramanujan_prime_lower(nlo) - 1;
maxk = nth_ramanujan_prime_upper(nhi) + 1;
if (mink < 15) mink = 15;
if (mink % 2 == 0) mink--;
if (verbose >= 2) { printf("Rn[%"UVuf"] to Rn[%"UVuf"] Noe's: %"UVuf" to %"UVuf"\n", nlo, nhi, mink, maxk); fflush(stdout); }
s = 1 + prime_count(2,mink-2) - prime_count(2,(mink-1)>>1);
{
unsigned char *segment, *seg2 = 0;
void* ctx = start_segment_primes(mink, maxk, &segment);
UV seg_base, seg_low, seg_high, new_size, seg2beg, seg2end, seg2size = 0;
while (next_segment_primes(ctx, &seg_base, &seg_low, &seg_high)) {
seg2beg = 30 * (((seg_low+1)>>1)/30);
seg2end = 30 * ((((seg_high+1)>>1)+29)/30);
new_size = (seg2end - seg2beg)/30 + 1;
if (new_size > seg2size) {
if (seg2size > 0) Safefree(seg2);
New(0, seg2, new_size, unsigned char);
seg2size = new_size;
}
(void) sieve_segment(seg2, seg2beg/30, seg2end/30);
for (k = seg_low; k <= seg_high; k += 2) {
if (is_prime_in_sieve(segment, k-seg_base)) s++;
if (s >= nlo && s <= nhi) L[s-nlo] = k+1;
if ((k & 3) == 1 && is_prime_in_sieve(seg2, ((k+1)>>1)-seg2beg)) s--;
if (s >= nlo && s <= nhi) L[s-nlo] = k+2;
}
}
end_segment_primes(ctx);
Safefree(seg2);
}
if (verbose >= 2) { printf("Generated %"UVuf" Ramanujan primes from %"UVuf" to %"UVuf"\n", nhi-nlo+1, L[0], L[nhi-nlo]); fflush(stdout); }
return L;
}
UV nth_ramanujan_prime(UV n) {
UV rn, *L;
if (n <= 2) return (n == 0) ? 0 : (n == 1) ? 2 : 11;
L = n_range_ramanujan_primes(n, n);
rn = L[0];
Safefree(L);
return rn;
}
/* Returns array of Ram primes between low and high, results from first->last */
UV* ramanujan_primes(UV* first, UV* last, UV low, UV high)
{
UV nlo, nhi, *L, lo, hi, mid;
if (high < 2 || high < low) return 0;
if (low < 2) low = 2;
nlo = ramanujan_prime_count_lower(low);
nhi = ramanujan_prime_count_upper(high);
L = n_range_ramanujan_primes(nlo, nhi);
/* Search for first entry in range */
for (lo = 0, hi = nhi-nlo+1; lo < hi; ) {
mid = lo + (hi-lo)/2;
if (L[mid] < low) lo = mid+1;
else hi = mid;
}
*first = lo;
/* Search for last entry in range */
for (hi = nhi-nlo+1; lo < hi; ) {
mid = lo + (hi-lo)/2;
if (L[mid] <= high) lo = mid+1;
else hi = mid;
}
*last = lo-1;
return L;
}
int is_ramanujan_prime(UV n) {
UV beg, end, *L;
if (!is_prime(n)) return 0;
if (n < 17) return (n == 2 || n == 11);
/* Generate Ramanujan primes and see if we're in the list. Slow. */
L = ramanujan_primes(&beg, &end, n, n);
Safefree(L);
return (beg <= end);
}
UV ramanujan_prime_count_approx(UV n)
{
/* Binary search on nth_ramanujan_prime_approx */
UV lo, hi;
if (n < 29) return (n < 2) ? 0 : (n < 11) ? 1 : (n < 17) ? 2 : 3;
lo = ramanujan_prime_count_lower(n);
hi = ramanujan_prime_count_upper(n);
while (lo < hi) {
UV mid = lo + (hi-lo)/2;
if (nth_ramanujan_prime_approx(mid) < n) lo = mid+1;
else hi = mid;
}
return lo-1;
}
UV nth_ramanujan_prime_approx(UV n)
{
UV lo = nth_ramanujan_prime_lower(n), hi = nth_ramanujan_prime_upper(n);
/* Our upper bounds come out much closer, so weight toward them. */
double weight = (n <= UVCONST(4294967295)) ? 1.62 : 1.51;
return lo + weight * ((hi-lo) >> 1);
}
#if BITS_PER_WORD == 64
#define RAMPC2 56
static const UV ramanujan_counts_pow2[RAMPC2+1] = {
0, 1, 1, 1, 2, 4, 7, 13, 23, 42, 75, 137, 255, 463, 872, 1612,
3030, 5706, 10749, 20387, 38635, 73584, 140336, 268216, 513705,
985818, 1894120, 3645744, 7027290, 13561906, 26207278, 50697533,
98182656, 190335585, 369323301, 717267167,
UVCONST( 1394192236), UVCONST( 2712103833), UVCONST( 5279763823),
UVCONST( 10285641777), UVCONST( 20051180846), UVCONST( 39113482639),
UVCONST( 76344462797), UVCONST( 149100679004), UVCONST( 291354668495),
UVCONST( 569630404447), UVCONST( 1114251967767), UVCONST( 2180634225768),
UVCONST( 4269555883751), UVCONST( 8363243713305), UVCONST( 16388947026629),
UVCONST( 32129520311897), UVCONST( 63012603695171), UVCONST(123627200537929),
UVCONST(242637500756376), UVCONST(476379740340417), UVCONST(935609435783647) };
#else
#define RAMPC2 31 /* input limited */
static const UV ramanujan_counts_pow2[RAMPC2+1] = {
0, 1, 1, 1, 2, 4, 7, 13, 23, 42, 75, 137, 255, 463, 872, 1612,
3030, 5706, 10749, 20387, 38635, 73584, 140336, 268216, 513705,
985818, 1894120, 3645744, 7027290, 13561906, 26207278, 50697533 };
#endif
static UV _ramanujan_prime_count(UV n) {
UV i, v, rn, *L, window, swin, ewin, wlen, log2 = log2floor(n), winmult = 1;
if (n <= 10) return (n < 2) ? 0 : 1;
/* We have some perfect powers of 2 in our table */
if ((n & (n-1)) == 0 && log2 <= RAMPC2)
return ramanujan_counts_pow2[log2];
if (_XS_get_verbose()) { printf("ramanujan_prime_count calculating Pi(%"UVuf")\n",n); fflush(stdout); }
v = prime_count(2,n) - prime_count(2,n >> 1);
/* For large enough n make a slightly bigger window */
if (n > 1000000000U) winmult = 16;
while (1) {
window = 20 * winmult;
swin = (v <= window) ? 1 : v-window;
ewin = v+window;
wlen = ewin-swin+1;
L = n_range_ramanujan_primes(swin, ewin);
if (L[0] < n && L[wlen-1] > n) {
/* Naive linear search from the start. */
for (i = 1; i < wlen; i++)
if (L[i] > n && L[i-1] <= n)
break;
if (i < wlen) break;
}
winmult *= 2;
if (_XS_get_verbose()) { printf(" ramanujan_prime_count increasing window\n"); fflush(stdout); }
}
rn = swin + i - 1;
Safefree(L);
return rn;
}
UV ramanujan_prime_count(UV lo, UV hi)
{
UV count;
if (hi < 2 || hi < lo) return 0;
#if 1
count = _ramanujan_prime_count(hi);
if (lo > 2)
count -= _ramanujan_prime_count(lo-1);
#else
{
UV beg, end, *L;
/* Generate all Rp from lo to hi */
L = ramanujan_primes(&beg, &end, lo, hi);
count = (L && end >= beg) ? end-beg+1 : 0;
Safefree(L);
}
#endif
return count;
}