ISSN 1088-6842(online) ISSN 0025-5718(print)

Finding prime pairs with particular gaps

Author: Pamela A. Cutter
Journal: Math. Comp. 70 (2001), 1737-1744
MSC (2000): Primary 11A41; Secondary 11Y11, 11Y55, 11Y99
DOI: https://doi.org/10.1090/S0025-5718-01-01327-8
Published electronically: May 11, 2001
MathSciNet review: 1836931
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By a prime gap of size , we mean that there are primes and such that the numbers between and are all composite. It is widely believed that infinitely many prime gaps of size exist for all even integers . However, it had not previously been known whether a prime gap of size existed. The objective of this article was to be the first to find a prime gap of size , by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from to , and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form , , and their application to divisibility of binomial coefficients by a square will also be discussed.

References [Enhancements On Off] (What's this?)

• 1. J. Brillhart, D.H. Lehmer, and J.L. Selfridge, New primality criteria and factorizations of , Math. Comp. 29:130 (1975), 620-647. MR 83j:10010
• 2. A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107. MR 97m:11023
• 3. G.H. Hardy and J.E. Littlewood, Some problems on partitio numerorum III. On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.
• 4. T. Nicely, New maximal prime gaps and first occurrences, Math. Comp, 68:227 (1999) 1311-1315. MR 99i:11004
• 5. P. Ribenboim, The new book of prime number records, Springer, New York, 1996. MR 96k:11112
• 6. D. Shanks, On maximal gaps between successive primes, Math. Comp. 18 (1964), 646-651. MR 29:4745
• 7. S. Weintraub, A prime gap of 864, J. Recreational Math 25:1 (1993), 42-43.
• 8. J. Young and A. Potler, First occurrence prime gaps, Math. Comp. 52:185 (1989), 221-224. MR 89f:11019

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