{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T05:41:01Z","timestamp":1696484461960},"reference-count":9,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2015,5,22]],"date-time":"2015-05-22T00:00:00Z","timestamp":1432252800000},"content-version":"vor","delay-in-days":21,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2015,5]]},"abstract":"<jats:p>In 1893, Schatunowsky showed that 30 is the largest number all of whose totatives are primes; we show that this result cannot be proved, in any form, in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201400064-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201400064:malq201400064-math-0001\" \/> Chebyshev's theorem (Bertrand's postulate), even if all irreducibles are primes. Bonse's inequality is shown to be indeed weaker than Chebyshev's theorem. Schatunowsky's theorem holds in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201400064-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201400064:malq201400064-math-0002\" \/> together with Bonse's inequality, the existence of the greatest prime dividing certain types of numbers, and the condition that all irreducibles be prime.<\/jats:p>","DOI":"10.1002\/malq.201400064","type":"journal-article","created":{"date-parts":[[2015,5,22]],"date-time":"2015-05-22T23:46:43Z","timestamp":1432338403000},"page":"230-235","source":"Crossref","is-referenced-by-count":1,"title":["Schatunowsky's theorem, Bonse's inequality, and Chebyshev's theorem in weak fragments of Peano arithmetic"],"prefix":"10.1002","volume":"61","author":[{"given":"Victor","family":"Pambuccian","sequence":"first","affiliation":[{"name":"School of Mathematical and Natural Sciences New College of Interdisciplinary Arts &amp; Sciences, Arizona State University, West Campus P. O. Box 37100 Phoenix AZ 85069\u20107100 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2015,5,22]]},"reference":[{"key":"e_1_2_8_2_1","first-page":"149","volume-title":"Lecture Notes in Logic Vol. 18","author":"D'Aquino P.","year":"2005"},{"issue":"3","key":"e_1_2_8_3_1","first-page":"292","article-title":"\u00dcber eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung","volume":"12","author":"Bonse H.","year":"1907","journal-title":"Arch. Math. Phys."},{"key":"e_1_2_8_4_1","volume-title":"Volume I: Divisibility and Primality","author":"Dickson L. E.","year":"1966"},{"key":"e_1_2_8_5_1","volume-title":"An Introduction, second edition","author":"Hungerford T. W.","year":"1997"},{"key":"e_1_2_8_6_1","volume-title":"Oxford Logic Guides Vol. 15","author":"Kaye R.","year":"1991"},{"key":"e_1_2_8_7_1","volume-title":"Algebra, revised third edition","author":"Lang S.","year":"2002"},{"key":"e_1_2_8_8_1","doi-asserted-by":"crossref","unstructured":"H.Rademacher O.Toeplitz Von Zahlen und Figuren: Proben Mathematischen Denkens f\u00fcr Liebhaber der Mathematik second edition (Springer 1933).","DOI":"10.1007\/978-3-662-36239-6"},{"key":"e_1_2_8_9_1","first-page":"79","article-title":"A non\u2010standard model for a free variable fragment of number theory","volume":"12","author":"Shepherdson J. C.","year":"1964","journal-title":"Bull. Acad. Pol. Sci. S\u00e9r. Sci. Math. Astronom. Phys."},{"key":"e_1_2_8_10_1","first-page":"285","volume-title":"Studies in Logic and the Foundations of Mathematics Vol. 96","author":"Wilkie A. J.","year":"1978"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201400064","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201400064","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T02:34:28Z","timestamp":1696386868000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201400064"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,5]]}},"alternative-id":["10.1002\/malq.201400064"],"URL":"https:\/\/doi.org\/10.1002\/malq.201400064","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,5]]}}}