Gap Correlations Among Prime Numbers

Eightomic reveals several non-trivial prime number gap pattern sequences from automated computational pattern analysis development.

Explanation

Mathematicians and scientists study prime numbers to this day relentlessly without discovering any significant non-random patterns.

While developing Coherence, an automated pattern recognition API, Eightomic noticed the following consistent pattern in prime numbers when any set of adjacent prime numbers span a gap sized at a multiple of 10.

In a set of 3 adjacent prime numbers a b c, and when a and c have the difference of 10, b is the result of incrementing by either 4 or 6.

The first occurrence of this pattern, starting from the first prime number in ascending order, is 19 23 29.

There are several other patterns, for example, in a set of adjacent prime numbers a b c d and when a and d have the difference of 10, the difference of b and c is always 2.

The first occurrence of this pattern, starting from the first prime number in ascending order, is 7 11 13 17.

The following prime number generator code validates this for the first 20k primes.

#include <stdio.h>

int main(void) {
  unsigned int primes[20000];
  unsigned int dividend = 3;
  unsigned int divisor;
  unsigned short gap_occurrences_count = 0;
  unsigned short i = 1;
  unsigned char is_prime;
  unsigned char is_pattern = 1;

  while (i < 20000) {
    divisor = 3;
    is_prime = 1;

    while (
      dividend != divisor &&
      is_prime == 1
    ) {
      if ((dividend % divisor) == 0) {
        is_prime = 0;
      }

      divisor += 2;
    }

    if (is_prime == 1) {
      primes[i] = divisor;
      i++;
    }

    dividend += 2;
  }

  while (i > 9) {
    i--;

    if ((primes[i] - 10) == primes[i - 2]) {
      gap_occurrences_count++;

      if (
        (primes[i - 1] + 4) != primes[i] &&
        (primes[i - 1] + 6) != primes[i]
      ) {
        is_pattern = 0;
      }
    }
  }

  if (is_pattern == 1) {
    printf(
      "The prime number pattern is consistent among %u gap occurences.",
      gap_occurrences_count
    );
  } else {
    printf(
      "The prime number pattern is inconsistent among %u gap occurences.",
      gap_occurrences_count
    );
  }

  return 0;
}

Of the 20k primes, there are 1120 occurrences of the aforementioned pattern.

The following is a visualization of the first 100 occurences and their increments.

19     +4   23     +6   29
31     +6   37     +4   41
43     +4   47     +6   53
61     +6   67     +4   71
73     +6   79     +4   83
79     +4   83     +6   89
127    +4   131    +6   137
157    +6   163    +4   167
163    +4   167    +6   173
229    +4   233    +6   239
271    +6   277    +4   281
349    +4   353    +6   359
373    +6   379    +4   383
379    +4   383    +6   389
433    +6   439    +4   443
439    +4   443    +6   449
499    +4   503    +6   509
607    +6   613    +4   617
643    +4   647    +6   653
673    +4   677    +6   683
733    +6   739    +4   743
751    +6   757    +4   761
937    +4   941    +6   947
967    +4   971    +6   977
1009   +4   1013   +6   1019
1093   +4   1097   +6   1103
1213   +4   1217   +6   1223
1279   +4   1283   +6   1289
1291   +6   1297   +4   1301
1429   +4   1433   +6   1439
1489   +4   1493   +6   1499
1543   +6   1549   +4   1553
1549   +4   1553   +6   1559
1597   +4   1601   +6   1607
1609   +4   1613   +6   1619
1657   +6   1663   +4   1667
1777   +6   1783   +4   1787
1861   +6   1867   +4   1871
1987   +6   1993   +4   1997
2131   +6   2137   +4   2141
2203   +4   2207   +6   2213
2287   +6   2293   +4   2297
2341   +6   2347   +4   2351
2347   +4   2351   +6   2357
2371   +6   2377   +4   2381
2383   +6   2389   +4   2393
2389   +4   2393   +6   2399
2437   +4   2441   +6   2447
2467   +6   2473   +4   2477
2539   +4   2543   +6   2549
2677   +6   2683   +4   2687
2689   +4   2693   +6   2699
2791   +6   2797   +4   2801
2833   +4   2837   +6   2843
2851   +6   2857   +4   2861
2953   +4   2957   +6   2963
3079   +4   3083   +6   3089
3181   +6   3187   +4   3191
3313   +6   3319   +4   3323
3319   +4   3323   +6   3329
3529   +4   3533   +6   3539
3607   +6   3613   +4   3617
3613   +4   3617   +6   3623
3691   +6   3697   +4   3701
3793   +4   3797   +6   3803
3907   +4   3911   +6   3917
3919   +4   3923   +6   3929
4003   +4   4007   +6   4013
4129   +4   4133   +6   4139
4441   +6   4447   +4   4451
4447   +4   4451   +6   4457
4507   +6   4513   +4   4517
4639   +4   4643   +6   4649
4723   +6   4729   +4   4733
4789   +4   4793   +6   4799
4933   +4   4937   +6   4943
4993   +6   4999   +4   5003
4999   +4   5003   +6   5009
5077   +4   5081   +6   5087
5407   +6   5413   +4   5417
5431   +6   5437   +4   5441
5521   +6   5527   +4   5531
5563   +6   5569   +4   5573
5641   +6   5647   +4   5651
5683   +6   5689   +4   5693
5839   +4   5843   +6   5849
5851   +6   5857   +4   5861
5857   +4   5861   +6   5867
6037   +6   6043   +4   6047
6043   +4   6047   +6   6053
6211   +6   6217   +4   6221
6571   +6   6577   +4   6581
6907   +4   6911   +6   6917
6961   +6   6967   +4   6971
6967   +4   6971   +6   6977
6991   +6   6997   +4   7001
7237   +6   7243   +4   7247
7243   +4   7247   +6   7253
7477   +4   7481   +6   7487
7537   +4   7541   +6   7547

The aforementioned pattern can directly contribute to further cryptanalytic proofs required by secure hashing algorithms that leverage the entropic security properties of prime number derivatives.

Additional sub-patterns could suggest the existence of undiscovered pattern occurences hidden in nature.

Further analysis may eventually reveal a deterministic prime number increment formula to revolutionize computational primality testing and redefine the constraining mysteries of mathematics and science.