{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T01:20:19Z","timestamp":1777425619270,"version":"3.51.4"},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2016,9,1]],"date-time":"2016-09-01T00:00:00Z","timestamp":1472688000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/legalcode"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p> Representation of a non zero integer as a signed product of primes is unique similarly to its representations in various types of positional notations [4], [3]. The study focuses on counting the prime factors of integers in the form of sums or differences of two equal powers (thus being represented by 1 and a series of zeroes in respective digital bases).<\/jats:p>\n               <jats:p>Although the introduced theorems are not particularly important, they provide a couple of shortcuts useful for integer factorization, which could serve in further development of Mizar projects [2]. This could be regarded as one of the important benefits of proof formalization [9].<\/jats:p>","DOI":"10.1515\/forma-2016-0015","type":"journal-article","created":{"date-parts":[[2017,2,14]],"date-time":"2017-02-14T10:02:03Z","timestamp":1487066523000},"page":"187-198","source":"Crossref","is-referenced-by-count":4,"title":["Prime Factorization of Sums and Differences of Two Like Powers"],"prefix":"10.1515","volume":"24","author":[{"given":"Rafa\u0142","family":"Ziobro","sequence":"first","affiliation":[{"name":"Department of Carbohydrate Technology University of Agriculture Krakow, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,2,21]]},"reference":[{"key":"2021040802414435984_j_forma-2016-0015_ref_1_w2aab2b8c10b1b7b1ab1ab1Aa","unstructured":"[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990."},{"key":"2021040802414435984_j_forma-2016-0015_ref_2_w2aab2b8c10b1b7b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Bylinski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, 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 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2021040802414435984_j_forma-2016-0015_ref_3_w2aab2b8c10b1b7b1ab1ab3Aa","doi-asserted-by":"crossref","unstructured":"[3] Paul Erd\u0151s and J\u00e1nos Sur\u00e1nyi. Topics in the Theory of Numbers, chapter Divisibility, the Fundamental Theorem of Number Theory, pages 1-37. Springer New York, 2003. doi:10.1007\/978-1-4613-0015-1 1.","DOI":"10.1007\/978-1-4613-0015-1"},{"key":"2021040802414435984_j_forma-2016-0015_ref_4_w2aab2b8c10b1b7b1ab1ab4Aa","unstructured":"[4] Jacek Gancarzewicz. Arytmetyka, 2000. In Polish."},{"key":"2021040802414435984_j_forma-2016-0015_ref_5_w2aab2b8c10b1b7b1ab1ab5Aa","unstructured":"[5] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573-577, 1997."},{"key":"2021040802414435984_j_forma-2016-0015_ref_6_w2aab2b8c10b1b7b1ab1ab6Aa","unstructured":"[6] Artur Korni\u0142owicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179-186, 2004."},{"key":"2021040802414435984_j_forma-2016-0015_ref_7_w2aab2b8c10b1b7b1ab1ab7Aa","unstructured":"[7] Rafa\u0142 Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990."},{"key":"2021040802414435984_j_forma-2016-0015_ref_8_w2aab2b8c10b1b7b1ab1ab8Aa","unstructured":"[8] Rafa\u0142 Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990."},{"key":"2021040802414435984_j_forma-2016-0015_ref_9_w2aab2b8c10b1b7b1ab1ab9Aa","doi-asserted-by":"crossref","unstructured":"[9] Adam Naumowicz. An example of formalizing recent mathematical results in Mizar. Journal of Applied Logic, 4(4):396-413, 2006. doi:10.1016\/j.jal.2005.10.003. Towards Computer Aided Mathematics.","DOI":"10.1016\/j.jal.2005.10.003"},{"key":"2021040802414435984_j_forma-2016-0015_ref_10_w2aab2b8c10b1b7b1ab1ac10Aa","unstructured":"[10] Akira Nishino and Yasunari Shidama. The Maclaurin expansions. Formalized Mathematics, 13(3):421-425, 2005."},{"key":"2021040802414435984_j_forma-2016-0015_ref_11_w2aab2b8c10b1b7b1ab1ac11Aa","unstructured":"[11] Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991."},{"key":"2021040802414435984_j_forma-2016-0015_ref_12_w2aab2b8c10b1b7b1ab1ac12Aa","doi-asserted-by":"crossref","unstructured":"[12] Marco Riccardi. Pocklington\u2019s theorem and Bertrand\u2019s postulate. Formalized Mathematics, 14(2):47-52, 2006. doi:10.2478\/v10037-006-0007-y.","DOI":"10.2478\/v10037-006-0007-y"},{"key":"2021040802414435984_j_forma-2016-0015_ref_13_w2aab2b8c10b1b7b1ab1ac13Aa","unstructured":"[13] Piotr Rudnicki and Andrzej Trybulec. Abian\u2019s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997."},{"key":"2021040802414435984_j_forma-2016-0015_ref_14_w2aab2b8c10b1b7b1ab1ac14Aa","doi-asserted-by":"crossref","unstructured":"[14] Rafa\u0142 Ziobro. Fermat\u2019s Little Theorem via divisibility of Newton\u2019s binomial. 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