First occurrence prime gaps

Abstract

Abstract.An ongoing search for first occurrence prime gaps continues.An ongoing search for first occurrence prime gaps is being carried out which extends all previous work done on this subject. To date this search has found all such gaps for primes up to 7.263 x 1013. First occurrence prime gaps had previously been known for primes less than 4.444 x 1012 [2]. Several gaps larger than the previously largest gap of 682 (not a first occurrence) found by Weintraub [4] have been found.Computer programs were written in FORTRAN and CAL (Cray Assembly Language) on a CRAY-2 supercomputer.The computation was conducted as follows. A sufficient number of primes were generated to perform a sieve. Odd numbers beginning with 3 were sieved one block at a time, where each block was chosen to contain 40,000,000 numbers based on system resource availability. The even numbers were eliminated during initialization of each block. One number was stored per 64-bit computer word. After sieving each block, the differences between consecutive primes were calculated and stored. This was accomplished by loading 64 elements of the sieved block at a time into a vector register. A 64-bit vector mask was created containing l's for corresponding nonzero values in the vector register. If the vector mask was zero, the next 64 values of the sieved block were loaded into the vector register. If the mask was nonzero, an instruction to count the number of leading zeros was executed to get the offset from the beginning of the vector register for the next prime. A subtraction of the previous prime was done, thus arriving at the gap. The leftmost 1 of the vector mask was cleared and the method repeated, beginning with checking if the vector mask was zero. This was repeated until the entire block was processed. The last prime in each block was saved in order to calculate the difference between that prime and the first prime in the next block to make sure no gaps were missed. The time to perform the sieve for each block, and to calculate all gaps generated, was about 10.5 seconds for numbers in the range of 7.2 x 1013. The largest prime in the last block processed to date is 72635119999997, so the table of first occurrence prime gaps is complete to that prime.

Iterated absolute values of differences of consecutive primes

“…Over 104 blocks of length 8 x 106 each around 5.25 x 1012 (see §4 for a definition), the correlation between maximal values of g(n) and maximal prime gaps in a block was 0.52. The largest value of g(n) that was found, g(n) = 635 for n close to 7r(7.17716 x 1012), is due to the prime gap 674, which is the largest one up to that height ( [3,20]). The second largest value of g(n), g(n) = 589 for n close to 7r(2.61494 x 1012), is due to the prime gap 652, which is also the largest one up to its height.…”

“…Maximal gaps between consecutive primes around x are thought to be not much larger than (logx)2. (There is a conjecture of Cramer [5] that these gaps are 0((logx)2), and numerical evidence [3,4,20] supports this conjecture as well as a slightly stronger one of Shanks [16]. There are heuristic arguments, based on work of Maier [9], that suggest the true order of magnitude might be slightly larger, but at most by some fractional power of log log x.)…”

Iterated Absolute Values of Differences of Consecutive Primes

“…Over 104 blocks of length 8 x 106 each around 5.25 x 1012 (see §4 for a definition), the correlation between maximal values of g(n) and maximal prime gaps in a block was 0.52. The largest value of g(n) that was found, g(n) = 635 for n close to 7r(7.17716 x 1012), is due to the prime gap 674, which is the largest one up to that height ( [3,20]). The second largest value of g(n), g(n) = 589 for n close to 7r(2.61494 x 1012), is due to the prime gap 652, which is also the largest one up to its height.…”

“…Maximal gaps between consecutive primes around x are thought to be not much larger than (logx)2. (There is a conjecture of Cramer [5] that these gaps are 0((logx)2), and numerical evidence [3,4,20] supports this conjecture as well as a slightly stronger one of Shanks [16]. There are heuristic arguments, based on work of Maier [9], that suggest the true order of magnitude might be slightly larger, but at most by some fractional power of log log x.)…”

Book Review: The book of prime number records

“…What then is the largest x so that all primes up to x have ever been formed and studied in a computer memory? Young and Potier [10] have computed all primes up to 7.2635 x 10 13 and studied the gaps between consecutive primes. Their record-making calculations are still continuing now.…”