[Paper Review] Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials

Vladimir Pletser|arXiv (Cornell University)|Sep 29, 2014

Advanced Mathematical Theories and Applications19 references3 citations

TL;DR

This paper presents a complete analytical method to find all integers $ s $ such that the sum of $ M $ consecutive squared integers starting from $ a^2 \geq 1 $ is a perfect square. It uses generalized Pell equations and Chebyshev polynomials to generate infinite families of solutions when $ M $ is not a square, and reduces the problem to a finite difference of squares when $ M $ is a square, yielding all possible solutions systematically.

ABSTRACT

Square roots $s$ of sums of $M$ consecutive integer squares starting from $a^{2}\geq1$ are integers if $M\equiv0,9,24$ or $33(mod\,72)$; or $M\equiv1,2$ or $16(mod\,24)$; or $M\equiv11(mod\,12)$ and cannot be integers if $M\equiv3,5,6,7,8$ or $10(mod\,12)$. Finding all solutions with $s$ integer requires to solve a Diophantine quadratic equation in variables $a$ and $s$ with $M$ as a parameter. If $M$ is not a square integer, the Diophantine quadratic equation in variables $a$ and $s$ is transformed into a generalized Pell equation whose form depends on the $M(mod\,4)$ congruent value, and whose solutions, if existing, yield all the solutions in $a$ and $s$ for a given value of $M$. Depending on whether this generalized Pell equation admits one or several fundamental solution(s), there are one or several infinite branches of solutions in $a$ and $s$ that can be written simply in function of Chebyshev polynomials evaluated at the fundamental solutions of the related simple Pell equation. If $M$ is a square integer, it is known that $M\equiv1(mod\,24)$ and $M=(6n-1)^{2}$ for all integers $n$; then the Diophantine quadratic equation in variables $a$ and $s$ reduces to a simple difference of integer squares which yields a finite number of solutions in $a$ and $s$ to the initial problem.

Motivation & Objective

• To systematically determine all integer solutions $ (a, s) $ such that the sum of $ M $ consecutive squares starting from $ a^2 \geq 1 $ equals $ s^2 $.
• To classify solutions based on whether $ M $ is a square or not, as this determines the underlying Diophantine structure.
• To provide a unified analytical framework using generalized Pell equations and Chebyshev polynomials for generating all solutions when $ M $ is not a square.
• To show that when $ M $ is a square, the problem reduces to a finite difference of squares, yielding only finitely many solutions.
• To extend prior work on Lucas’ cannonball problem by characterizing all possible $ M $ and corresponding $ a, s $ for which the sum of $ M $ consecutive squares is a perfect square.

Proposed method

• Transform the Diophantine equation $ \sum_{k=0}^{M-1} (a+k)^2 = s^2 $ into a quadratic Diophantine equation in $ a $ and $ s $, parameterized by $ M $.
• For non-square $ M $, reduce the equation to a generalized Pell equation whose form depends on $ M \mod 4 $.
• Use the fundamental solutions of the associated simple Pell equation to generate infinite solution branches via Chebyshev polynomials.
• For square $ M $, express the equation as a difference of integer squares, leading to a finite number of factorizations to solve.
• Apply the identity $ X^2 - Y^2 = N $ with even factor pairs to enumerate all valid $ (X, Y) $, and recover $ s $ and $ a $ from $ X = s/(6n-1) $, $ Y = a + 6n(3n-1) $ when $ M = (6n-1)^2 $.
• Filter solutions to exclude cases where $ a < 0 $, ensuring $ a \geq 1 $ as required.

Experimental results

Research questions

• RQ1 For which values of $ M $ does the sum of $ M $ consecutive squared integers equal a perfect square?
• RQ2 How can all integer solutions $ (a, s) $ be systematically generated for a given $ M $, especially when $ M $ is not a square?
• RQ3 What is the role of Chebyshev polynomials in parameterizing the infinite solution branches of the generalized Pell equation arising from this problem?
• RQ4 Why does the problem reduce to a finite number of solutions when $ M $ is a perfect square, and how can these be enumerated?
• RQ5 How do modular constraints on $ M $, such as $ M \equiv 0,9,24,33 \mod 72 $, determine the existence of solutions?

Key findings

• Integer solutions exist only if $ M \equiv 0,9,24,33 \mod 72 $, or $ M \equiv 1,2,16 \mod 24 $, or $ M \equiv 11 \mod 12 $.
• When $ M $ is not a square, all solutions in $ a $ and $ s $ are generated from fundamental solutions of a generalized Pell equation using Chebyshev polynomials evaluated at the fundamental solution of the related simple Pell equation.
• For $ M = 25 $, the only solution is $ a = 0, s = 70 $, which is invalid for $ a \geq 1 $, but corresponds to the known solution for $ M = 24 $.
• For $ M = 289 = (6 \cdot 3 - 1)^2 $, there are 12 factor pairs of $ N = 6960 $, but only 7 yield $ a_k \geq 1 $, giving 7 valid solutions with $ s_k $ ranging from 3128 to 29597.
• When $ M $ is a square, $ M = (6n-1)^2 $, the number of solutions is finite and determined by the number of even factorizations of $ N = 2n(3n-1)(6n(3n-1)+1) $.
• The method provides a complete and analytical classification of all solutions, with explicit formulas for $ a_k $ and $ s_k $ in terms of factor pairs and Chebyshev polynomials.