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 ∑j=02m−1(A+j)3 = ∑j=0m−1(B+2j)3. An explicit infinite family is obtained from the Pell-type equation 48329z2−156m2=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.
Related
- Long runs of integers with small prime factors and the divisor function of n!
- Long runs and rare patterns of a random completely multiplicative function
- Initial Runs in Integer Compositions: Lambert Series and Generalized Divisor Sums
- The density of long runs of non-r-free integers
- On concatenations of two k-generalized Pell numbers
BibTeX
@misc{eydelzon2026from,
title = {{From Pythagorean Runs to Pell-Generated Cubic Runs}},
author = {Anatoly Eydelzon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09714},
eprint = {2609.09714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09714},
}
RIS
TY - UNPB
TI - From Pythagorean Runs to Pell-Generated Cubic Runs
AU - Anatoly Eydelzon
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.09714
UR - https://arxiv.org/abs/2609.09714
AB - 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 ∑_j=0^2m−1(A+j)^3 = ∑_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.
ER -