General solutions of sums of consecutive cubed integers equal to squared integers

2015/01/24 by Vladimir Pletser, Pletser, Vladimir
Mathematics · #11D09 #11E25 #33D45 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D09 #msc:11E25 #msc:33D45

paper · pdf · doi:10.48550/arxiv.1501.06098

19 pages

arxiv created 2015/01/24 · openalex publication_date 2015/01/24 · arxiv updated 2015/01/27 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

All integer solutions (M,a,c) to the problem of the sums of M consecutive cubed integers (a+i)³ (a>1, 0≤ i≤ M-1) equaling squared integers c² are found by decomposing the product of the difference and sum of the triangular numbers of (a+M-1) and (a-1) in the product of their greatest common divisor g and remaining square factors δ² and σ², yielding c=gδσ. Further, the condition that g must be integer for several particular and general cases yield generalized Pell equations whose solutions allow to find all integer solutions (M,a,c) showing that these solutions appear recurrently. In particular, it is found that there always exist at least one solution for the cases of all odd values of M, of all odd integer square values of a, and of all even values of M equal to twice an integer square.

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