File: 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 */
  MPUverbose(2, "sieving to %"UVuf" for first %"UVuf" Ramanujan primes\n", max, n);
  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;

  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--;
  MPUverbose(2, "Rn[%"UVuf"] to Rn[%"UVuf"]     Noe's: %"UVuf" to %"UVuf"\n", nlo, nhi, mink, maxk);

  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);
  }
  MPUverbose(2, "Generated %"UVuf" Ramanujan primes from %"UVuf" to %"UVuf"\n", nhi-nlo+1, L[0], L[nhi-nlo]);
  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];

  MPUverbose(1, "ramanujan_prime_count calculating Pi(%"UVuf")\n",n);
  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;
    MPUverbose(1, "  ramanujan_prime_count increasing window\n");
  }
  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;
}