{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T01:29:29Z","timestamp":1747963769652},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2002,5,11]],"date-time":"2002-05-11T00:00:00Z","timestamp":1021075200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>By a prime gap of size <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n  <mml:semantics>\n    <mml:mi>g<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, we mean that there are primes <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n  <mml:semantics>\n    <mml:mi>p<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p plus g\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>g<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">p+g<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> such that the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g minus 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>g<\/mml:mi>\n      <mml:mo>\u2212<!-- \u2212 --><\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">g-1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> numbers between <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n  <mml:semantics>\n    <mml:mi>p<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p plus g\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>p<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>g<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">p+g<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> are all composite. It is widely believed that infinitely many prime gaps of size <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n  <mml:semantics>\n    <mml:mi>g<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> exist for all even integers <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n  <mml:semantics>\n    <mml:mi>g<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. However, it had not previously been known whether a prime gap of size <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1000\">\n  <mml:semantics>\n    <mml:mn>1000<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">1000<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> existed. The objective of this article was to be the first to find a prime gap of size <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1000\">\n  <mml:semantics>\n    <mml:mn>1000<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">1000<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, 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 <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"746\">\n  <mml:semantics>\n    <mml:mn>746<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">746<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> to <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1000\">\n  <mml:semantics>\n    <mml:mn>1000<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">1000<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, 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 <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1000\">\n  <mml:semantics>\n    <mml:mn>1000<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">1000<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> 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 <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1000\">\n  <mml:semantics>\n    <mml:mn>1000<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">1000<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"6 m plus 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>6<\/mml:mn>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">6m+1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"12 m minus 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>12<\/mml:mn>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>\u2212<!-- \u2212 --><\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">12m-1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"12 m plus 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>12<\/mml:mn>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">12m+1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and their application to divisibility of binomial coefficients by a square will also be discussed.<\/p>","DOI":"10.1090\/s0025-5718-01-01327-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1737-1744","source":"Crossref","is-referenced-by-count":3,"title":["Finding prime pairs with particular gaps"],"prefix":"10.1090","volume":"70","author":[{"given":"Pamela","family":"Cutter","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,5,11]]},"reference":[{"issue":"160","key":"1","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1090\/S0025-5718-1982-0669665-0","article-title":"Table errata: \u201cNew primality criteria and factorizations of 2^{\ud835\udc5a}\u00b11\u201d [Math. Comp. 29 (1975), 620\u2013647; MR 52 #5546] by the author, D. H. Lehmer and J. L. Selfridge","volume":"39","author":"Brillhart, John","year":"1982","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1112\/S0025579300011608","article-title":"Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients","volume":"43","author":"Granville, Andrew","year":"1996","journal-title":"Mathematika","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5793","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"G.H. Hardy and J.E. Littlewood, Some problems on partitio numerorum III. 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