An arbitrary number of squares whose sum, on excluding any one of them, is also a square

2025/05/03 by Ajai Choudhry, Choudhry, Ajai
Computer Science · Engineering · #11D09 #FOS: Mathematics #Finite set #Graph Labeling and Dimension Problems #Least-squares function approximation #Non-linear least squares #Number Theory (math.NT) #Parametric equation #Parametric statistics #Residual sum of squares #Square (algebra) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2505.01667

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

This paper is concerned with the problem of finding n distinct squares such that, on excluding any one of them, the sum of the remaining n-1 squares is a square. While parametric solutions are known when n=3 and n=4, when n > 4, only a finite number of numerical solutions, found by computer trials, are known. In fact, efforts to find parametric solutions for n > 4 have so far been futile. In this paper we describe two methods of obtaining parametric solutions of the problem, and we apply these methods to get several parametric solutions when n=5, 6, 7 or 8. We also indicate how parametric solutions may be obtained for larger values of n.

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