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.

The Little Book of Bigger Primes

“…The previous records were 1183 (in 2002) and 1131 (in 1999), both by Nyman. Other largest maximal gaps had been 803, by Young & Potler (1989), and 651, by Brent (1973).…”

Die Welt der Primzahlen

“…In der Reihenfolge ihres Erscheinens: , Brent (1973Brent ( , 1980, Young & Potler (1989) und Nicely (1999). In der Reihenfolge ihres Erscheinens: , Brent (1973Brent ( , 1980, Young & Potler (1989) und Nicely (1999).…”

“…In der Reihenfolge ihres Erscheinens: , Brent (1973Brent ( , 1980, Young & Potler (1989) und Nicely (1999). Frühere maximale Lücken hatten m = 803, gefunden von Young & Potler (1989), sowie m = 651, von Brent (1973 Wenn man die auf eine Primzahl p folgende Lücke g(p) bestimmt hat (die nicht notwendigerweise erstmalig auftreten muss), liegt es nahe sich zu fragen, inwieweit diese Lücke im Vergleich zu Primzahlen gleicher Größenordnung " ungewöhnlich groß" ist. Im August 2010 waren alle Primzahlen bis N = 1,6 × 10 18 untersucht.…”