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

Ajai Choudhry

TL;DR

The paper tackles the problem of finding $n$ distinct squares $igl"{x_i^2}$igr o$ such that excluding any one yields a remaining sum that is also a square. It develops two constructive, parametric approaches based on Diophantine chains and sum-of-two-squares identities to generate families of solutions for $n=5$–$8$, with explicit polynomial parameterizations and numerical examples. The results include high-degree constructions (up to degree $162$) via sign-alternation and a lemma, as well as lower-degree families (down to degree $38$ base, up to degree $114$ after refinement) from a Diophantine-chain framework. These methods extend the landscape of parametric solutions beyond the classical $n=3,4$ cases and provide a systematic path toward larger $n$, highlighting the trade-off between computational complexity and compact parametrizations.

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 $n$.

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

TL;DR

The paper tackles the problem of finding distinct squares igr on=5816238114n=3,4n$, highlighting the trade-off between computational complexity and compact parametrizations.

Abstract

This paper is concerned with the problem of finding distinct squares such that, on excluding any one of them, the sum of the remaining squares is a square. While parametric solutions are known when and , when , only a finite number of numerical solutions, found by computer trials, are known. In fact, efforts to find parametric solutions for 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 or . We also indicate how parametric solutions may be obtained for larger values of .
Paper Structure (15 sections, 5 theorems, 55 equations)

This paper contains 15 sections, 5 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Two general methods for obtaining solutions in distinct integers of the simultaneous diophantine equations \ref{['diophchn']} and \ref{['csum']}
  4. First method
  5. Second method
  6. $n$ squares whose sum, on excluding any one of them is a square
  7. Five squares such that the sum of any four is a square
  8. Parametric solutions obtained by the first method
  9. Parametric solutions obtained by the second method
  10. Six squares such that the sum of any five is a square
  11. Parametric solutions obtained by the first method
  12. Parametric solutions obtained by the second method
  13. Seven squares such that the sum of any six is a square
  14. Eight squares such that the sum of any seven is a square
  15. Concluding remarks

Key Result

Lemma 1

There exist $n$ perfect squares $x_i^2, i=1, \ldots, n$, such that the sum of any $n-1$ of them is a perfect square if and only if the simultaneous diophantine equations, have a solution in integers.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 1
  • proof
  • Lemma 4
  • proof