{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T02:08:51Z","timestamp":1783908531516,"version":"3.55.0"},"reference-count":6,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,4,1]]},"abstract":"Summary<\/jats:title>\n \n In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form\n p<\/jats:italic>\n 2<\/jats:sup>\n + 1 =\n q<\/jats:italic>\n 2<\/jats:sup>\n +\n r<\/jats:italic>\n 2<\/jats:sup>\n , where\n p<\/jats:italic>\n ,\n q<\/jats:italic>\n ,\n r<\/jats:italic>\n are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers\n k<\/jats:italic>\n greater than zero such that all numbers of the form 2\n 2<\/jats:sup>\n \n n<\/jats:sup>\n <\/jats:italic>\n +\n k<\/jats:italic>\n (\n n<\/jats:italic>\n = 1, 2, . . . ) are composite.\n <\/jats:p>","DOI":"10.2478\/forma-2021-0006","type":"journal-article","created":{"date-parts":[[2021,8,27]],"date-time":"2021-08-27T11:48:21Z","timestamp":1630064901000},"page":"63-68","source":"Crossref","is-referenced-by-count":5,"title":["Elementary Number Theory Problems. Part II"],"prefix":"10.2478","volume":"29","author":[{"given":"Artur","family":"Korni\u0142owicz","sequence":"first","affiliation":[{"name":"Institute of Informatics , University of Bia\u0142ystok , Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dariusz","family":"Surowik","sequence":"additional","affiliation":[{"name":"University of Bia\u0142ystok , Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2021,8,26]]},"reference":[{"key":"2026071212491110025_j_forma-2021-0006_ref_001","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.10.1007\/978-3-319-20615-8_17","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026071212491110025_j_forma-2021-0006_ref_002","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026071212491110025_j_forma-2021-0006_ref_003","doi-asserted-by":"crossref","unstructured":"[3] Artur Korni\u0142owicz. Flexary connectives in Mizar. Computer Languages, Systems & Structures, 44:238\u2013250, December 2015. doi:10.1016\/j.cl.2015.07.002.10.1016\/j.cl.2015.07.002","DOI":"10.1016\/j.cl.2015.07.002"},{"key":"2026071212491110025_j_forma-2021-0006_ref_004","doi-asserted-by":"crossref","unstructured":"[4] Marco Riccardi. Pocklington\u2019s theorem and Bertrand\u2019s postulate. Formalized Mathematics, 14(2):47\u201352, 2006. doi:10.2478\/v10037-006-0007-y.10.2478\/v10037-006-0007-y","DOI":"10.2478\/v10037-006-0007-y"},{"key":"2026071212491110025_j_forma-2021-0006_ref_005","doi-asserted-by":"crossref","unstructured":"[5] Marco Riccardi. Solution of cubic and quartic equations. Formalized Mathematics, 17(2): 117\u2013122, 2009. doi:10.2478\/v10037-009-0012-z.10.2478\/v10037-009-0012-z","DOI":"10.2478\/v10037-009-0012-z"},{"key":"2026071212491110025_j_forma-2021-0006_ref_006","unstructured":"[6] Wac\u0142aw Sierpi\u0144ski. 250 Problems in Elementary Number Theory. Elsevier, 1970."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2021-0006","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T01:46:38Z","timestamp":1783907198000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2021-0006"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,1]]},"references-count":6,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,8,26]]},"published-print":{"date-parts":[[2021,4,1]]}},"alternative-id":["10.2478\/forma-2021-0006"],"URL":"https:\/\/doi.org\/10.2478\/forma-2021-0006","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,1]]}}}