{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T06:09:18Z","timestamp":1648879758479},"reference-count":7,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2002,3,22]],"date-time":"2002-03-22T00:00:00Z","timestamp":1016755200000},"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>A sequence of integers in an interval of length <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n  <mml:semantics>\n    <mml:mi>x<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is called admissible if for each prime there is a residue class modulo the prime which contains no elements of the sequence. The maximum number of elements in an admissible sequence in an interval of length <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n  <mml:semantics>\n    <mml:mi>x<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is denoted by <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho Superscript asterisk Baseline left-parenthesis x right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>\u03f1<!-- \u03f1 --><\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\varrho ^{*}(x)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Hensley and Richards showed that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho Superscript asterisk Baseline left-parenthesis x right-parenthesis greater-than pi left-parenthesis x right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>\u03f1<!-- \u03f1 --><\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mi>\u03c0<!-- \u03c0 --><\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\varrho ^{*}(x)&gt;\\pi (x)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> for large enough <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n  <mml:semantics>\n    <mml:mi>x<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We increase the known bounds on the set of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n  <mml:semantics>\n    <mml:mi>x<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> satisfying <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho Superscript asterisk Baseline left-parenthesis x right-parenthesis less-than-or-equal-to pi left-parenthesis x right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>\u03f1<!-- \u03f1 --><\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>\u2264<!-- \u2264 --><\/mml:mo>\n      <mml:mi>\u03c0<!-- \u03c0 --><\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\varrho ^{*}(x)\\le \\pi (x)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and find smaller values of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n  <mml:semantics>\n    <mml:mi>x<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> such that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho Superscript asterisk Baseline left-parenthesis x right-parenthesis greater-than pi left-parenthesis x right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>\u03f1<!-- \u03f1 --><\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mi>\u03c0<!-- \u03c0 --><\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\varrho ^{*}(x)&gt;\\pi (x)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We also find values of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n  <mml:semantics>\n    <mml:mi>x<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> satisfying <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho Superscript asterisk Baseline left-parenthesis x right-parenthesis greater-than 2 pi left-parenthesis x slash 2 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>\u03f1<!-- \u03f1 --><\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo>\u2217<!-- \u2217 --><\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mn>2<\/mml:mn>\n      <mml:mi>\u03c0<!-- \u03c0 --><\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mn>2<\/mml:mn>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\varrho ^{*}(x)&gt;2\\pi (x\/2)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. This shows the incompatibility of the conjecture <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"pi left-parenthesis x plus y right-parenthesis minus pi left-parenthesis y right-parenthesis less-than-or-equal-to 2 pi left-parenthesis x slash 2 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>\u03c0<!-- \u03c0 --><\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>y<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>\u2212<!-- \u2212 --><\/mml:mo>\n      <mml:mi>\u03c0<!-- \u03c0 --><\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>y<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>\u2264<!-- \u2264 --><\/mml:mo>\n      <mml:mn>2<\/mml:mn>\n      <mml:mi>\u03c0<!-- \u03c0 --><\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>x<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mn>2<\/mml:mn>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\pi (x+y)-\\pi (y)\\le 2\\pi (x\/2)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> with the prime <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n  <mml:semantics>\n    <mml:mi>k<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-tuples conjecture.<\/p>","DOI":"10.1090\/s0025-5718-01-01348-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1713-1718","source":"Crossref","is-referenced-by-count":3,"title":["Dense admissible sequences"],"prefix":"10.1090","volume":"70","author":[{"given":"David","family":"Clark","sequence":"first","affiliation":[]},{"given":"Norman","family":"Jarvis","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,3,22]]},"reference":[{"key":"1","first-page":"1","article-title":"Problems and results in number theory","author":"Erd\u0151s, P.","year":"1981"},{"key":"2","doi-asserted-by":"crossref","unstructured":"G.H. Hardy and J.E. Littlewood, Some problems of \u2019partitio numerorum\u2019. III. On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1\u201370.","DOI":"10.1007\/BF02403921"},{"key":"3","doi-asserted-by":"publisher","first-page":"375","DOI":"10.4064\/aa-25-4-375-391","article-title":"Primes in intervals","volume":"25","author":"Hensley, Douglas","year":"1973","journal-title":"Acta Arith.","ISSN":"http:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"4","series-title":"Progress in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0251-6","volume-title":"Prime numbers and computer methods for factorization","volume":"126","author":"Riesel, Hans","year":"1994","ISBN":"http:\/\/id.crossref.org\/isbn\/0817637435","edition":"2"},{"key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4064\/aa-7-1-1-8","article-title":"Remarks on the paper \u201cSur certaines hypoth\u00e8ses concernant les nombres premiers\u201d","volume":"7","author":"Schinzel, A.","year":"1961","journal-title":"Acta Arith.","ISSN":"http:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","first-page":"249","DOI":"10.2307\/2001943","article-title":"On a generalization of the prime pair problem","volume":"11","author":"Smith, Herschel F.","year":"1957","journal-title":"Math. Tables Aids Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/0891-6837","issn-type":"print"},{"key":"7","unstructured":"T. Vehka and I. Richards, Explicit construction of an admissible set for the conjecture that sometimes \ud835\udf0b(\ud835\udc65+\ud835\udc66)>\ud835\udf0b(\ud835\udc65)+\ud835\udf0b(\ud835\udc66), Notices Amer. Math. Soc. (1979), A\u2013453."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01348-5\/S0025-5718-01-01348-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01348-5\/S0025-5718-01-01348-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:39:06Z","timestamp":1627605546000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-01-01348-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,3,22]]},"references-count":7,"journal-issue":{"issue":"236","published-print":{"date-parts":[[2001,10]]}},"alternative-id":["S0025-5718-01-01348-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01348-5","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":[[2001,3,22]]}}}