# Needed: gaps between large primes

Edwin Clark eclark at math.usf.edu
Mon Nov 3 17:23:04 CET 2003

```Here's a list of 100 successive primes with 150 digits each and the gaps
between them. I also provide the Maple program used to generate them. If
you have more patience it would be easy to obtain such lists with even
larger primes. I first generated a random 150 digit number M and then find
the first 100 values of k such that M+k is prime. This only took a few
seconds. Of course, the primes are verified using Maple's procedure
isprime, which is a probabilistic primality test--but so far no
counter-examples have been found for it.

> num_digits:=150:
> M:=rand(10^(num_digits-1)..10^num_digits-1)();

M := 78947457415751382515065090500755340874818208281598492935963\
2269852681585809504709739738485231104248045693804710098188\
302655538010818866476054310788175

> L:=NULL:
> for k from 1   do
> if isprime(M+k) then L:=L,k;
> fi;
> if nops([L])= 100 then break;fi;
> od:
>
This is the list of k such that M+k is prime. For example M+164 and M+252
are prime.

> seq(L[i],i=1..100);

164, 252, 518, 672, 722, 1118, 1572, 1926, 1932, 2168, 2688, 2846,

3012, 3176, 3396, 3974, 4062, 4146, 4592, 4874, 5108, 5852,

5982, 7088, 7122, 7386, 7604, 7898, 7964, 7988, 8844, 8874,

9248, 9324, 9668, 9776, 9996, 10022, 10032, 10656, 10728,

10992, 11466, 11652, 11966, 12092, 12206, 12324, 12576, 12666,

13356, 13736, 14304, 15696, 15842, 15932, 16278, 16518, 16742,

17328, 17508, 18104, 18246, 18614, 19092, 19302, 19856, 19944,

20942, 21954, 22416, 22632, 22734, 23192, 23462, 23556, 24512,

24738, 25062, 25898, 25926, 26184, 26202, 26598, 26658, 27102,

27426, 27494, 27714, 28436, 29184, 29522, 31074, 31368, 31502,

31788, 32222, 32312, 32718, 32784

The gaps between successive values of the primes M+k are as follows:

> seq(L[i]-L[i-1],i=2..100);

88, 266, 154, 50, 396, 454, 354, 6, 236, 520, 158, 166, 164, 220,

578, 88, 84, 446, 282, 234, 744, 130, 1106, 34, 264, 218, 294,

66, 24, 856, 30, 374, 76, 344, 108, 220, 26, 10, 624, 72, 264,

474, 186, 314, 126, 114, 118, 252, 90, 690, 380, 568, 1392,

146, 90, 346, 240, 224, 586, 180, 596, 142, 368, 478, 210,

554, 88, 998, 1012, 462, 216, 102, 458, 270, 94, 956, 226,

324, 836, 28, 258, 18, 396, 60, 444, 324, 68, 220, 722, 748,

338, 1552, 294, 134, 286, 434, 90, 406, 66

Edwin

------------------------------------------------------------
W. Edwin Clark, Math Dept, University of South Florida,
http://www.math.usf.edu/~eclark/
------------------------------------------------------------

```