Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

2014/09/28 by Vladimir Pletser, Pletser, Vladimir
Mathematics · #11A07 #11E25 #Advanced Mathematical Identities #Analytic Number Theory Research #Benford’s Law and Fraud Detection #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1409.7969

openalex publication_date 2014/09/28 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Considering the problem of finding all the integer solutions of the sum of M consecutive integer squares starting at a² being equal to a squared integer s², it is shown that this problem has no solutions if M≡3,5,6,7,8 or 10 (mod 12) and has integer solutions if M≡0,9,24 or 33 (mod 72); or M≡1,2 or 16 (mod 24); or M≡11 (mod 12). All the allowed values of M are characterized using necessary conditions. If M is a square itself, then M≡1 (mod 24) and (M-1)/24 are all pentagonal numbers, except the first two.

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