{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T05:45:58Z","timestamp":1776923158329,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2001,7,18]],"date-time":"2001-07-18T00:00:00Z","timestamp":995414400000},"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>Using a carefully optimized segmented sieve and an efficient checking algorithm, the Goldbach conjecture has been verified and is now known to be true up to <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 dot 10 Superscript 14\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>4<\/mml:mn>\n      <mml:mo>\u22c5<\/mml:mo>\n      <mml:msup>\n        <mml:mn>10<\/mml:mn>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mn>14<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">4\\cdot 10^{14}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. The program was distributed to various workstations. It kept track of maximal values of the smaller prime <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> in the minimal partition of the even numbers, where a minimal partition is a representation <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 n equals p plus q\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>2<\/mml:mn>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>=<\/mml:mo>\n      <mml:mi>p<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>q<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">2n = p + q<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> with <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 n minus p prime\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>2<\/mml:mn>\n      <mml:mi>n<\/mml:mi>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:msup>\n        <mml:mi>p<\/mml:mi>\n        <mml:mo>\u2032<\/mml:mo>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">2n - p\u2019<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> being composite for all <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p prime greater-than p\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>p<\/mml:mi>\n        <mml:mo>\u2032<\/mml:mo>\n      <\/mml:msup>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mi>p<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">p\u2019 &gt; p<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. The maximal prime <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> needed in the considered interval was found to be 5569 and is needed for the partition 389965026819938 = 5569 + 389965026814369.<\/p>","DOI":"10.1090\/s0025-5718-00-01290-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1745-1749","source":"Crossref","is-referenced-by-count":27,"title":["Verifying the Goldbach conjecture up to 4\u22c510\u00b9\u2074"],"prefix":"10.1090","volume":"70","author":[{"given":"J\u00f6rg","family":"Richstein","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,7,18]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/bf01932283","article-title":"The segmented sieve of Eratosthenes and primes in arithmetic progressions to 10\u00b9\u00b2","volume":"17","author":"Bays, Carter","year":"1977","journal-title":"Nordisk Tidskr. Informationsbehandling (BIT)","ISSN":"https:\/\/id.crossref.org\/issn\/0901-246X","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/bf01933655","article-title":"Numerical results on the Goldbach conjecture","volume":"15","author":"Bohman, Jan","year":"1975","journal-title":"Nordisk Tidskr. Informationsbehandling (BIT)","ISSN":"https:\/\/id.crossref.org\/issn\/0901-246X","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"959","DOI":"10.2307\/2005532","article-title":"The first occurrence of large gaps between successive primes","volume":"27","author":"Brent, Richard P.","year":"1973","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"5","key":"4","first-page":"702","article-title":"On the Goldbach problem","volume":"32","author":"Chen, Jing Run","year":"1989","journal-title":"Acta Math. Sinica","ISSN":"https:\/\/id.crossref.org\/issn\/0583-1431","issn-type":"print"},{"key":"5","doi-asserted-by":"crossref","unstructured":"J. M. Deshouillers, H. J. J. te Riele, Y. Saouter New experimental results concerning the Goldbach conjecture, Proc. 3rd Int. Symp. on Algorithmic Number Theory, LNCS 1423 (1998), 204\u2013215.","DOI":"10.1007\/BFb0054863"},{"key":"6","unstructured":"P. H. Fuss, Correspondance math\u00e9matique et physique de quelques c\u00e9l\u00e8bres g\u00e9om\u00e8tres du XVIII^{\ud835\udc52} si\u00e8cle, tome I, St. P\u00e9tersbourg, (1843), 127+135."},{"key":"7","first-page":"423","article-title":"Checking the Goldbach conjecture on a vector computer","author":"Granville, A.","year":"1989"},{"issue":"204","key":"8","doi-asserted-by":"publisher","first-page":"915","DOI":"10.2307\/2153262","article-title":"On strong pseudoprimes to several bases","volume":"61","author":"Jaeschke, Gerhard","year":"1993","journal-title":"Math. 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Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"12","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/bf01939515","article-title":"On checking the Goldbach conjecture","volume":"4","author":"Shen, Mok-kong","year":"1964","journal-title":"Nordisk Tidskr. Informationsbehandling (BIT)","ISSN":"https:\/\/id.crossref.org\/issn\/0901-246X","issn-type":"print"},{"issue":"204","key":"13","doi-asserted-by":"publisher","first-page":"931","DOI":"10.2307\/2153264","article-title":"Checking the Goldbach conjecture up to 4\u22c510\u00b9\u00b9","volume":"61","author":"Sinisalo, Matti K.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"14","doi-asserted-by":"crossref","unstructured":"M. L. Stein, P. R. Stein, Experimental results on additive 2 bases, BIT 38 (1965), 427\u2013434.","DOI":"10.2307\/2688425"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01290-4\/S0025-5718-00-01290-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01290-4\/S0025-5718-00-01290-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:39:26Z","timestamp":1627605566000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01290-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,7,18]]},"references-count":14,"journal-issue":{"issue":"236","published-print":{"date-parts":[[2001,10]]}},"alternative-id":["S0025-5718-00-01290-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01290-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,7,18]]}}}