Abstract
Let p be a prime. We define S(p) the smallest number k such that every positive integer is a sum of at most k squares of integers that are not divisible by p. In this article, we prove that \(S(2)=10\), \(S(3)=6\), \(S(5)=5\), and \(S(p)=4\) for any prime p greater than 5. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by 5, except the unique positive integer 79.
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K. Kim supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2016R1A5A1008055 and NRF-2018R1C1B6007778) B.-K. Oh supported by the National Research Foundation of Korea (NRF-2019R1A2C1086347 and NRF-2020R1A5A1016126)
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Kim, K., Oh, BK. A sum of squares not divisible by a prime. Ramanujan J 59, 653–670 (2022). https://doi.org/10.1007/s11139-022-00591-3
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DOI: https://doi.org/10.1007/s11139-022-00591-3