From Pythagorean Runs to Pell-Generated Cubic Runs

Abstract

A classical Pythagorean run is an identity in which a block of consecutive squares is equal to the immediately following block of consecutive squares. Boardman's construction gives such a run for every prescribed length. We consider a cubic analogue in which the second block is allowed to have common difference 2. We prove that there are infinitely many positive integer triples \((m,A,B)\), with \(B>A+2m-1\), such that $$\sum _{j=0}^{2m-1}(A+j{)}^3=\sum _{j=0}^{m-1}(B+2j{)}^3.$$ An explicit infinite family is obtained from the Pell-type equation $$48329z^2-156m^2=161,$$ which reduces to a generalized Pell equation in normalized form. The ordering condition ensures that the step-2 progression begins strictly after the consecutive block ends. The resulting sequence of half-lengths satisfies an explicit second-order linear recurrence.