Skip to content
DraconiticResearch
Search Sources API
185,139 studies31 sourcesUpdated 34 min ago
PreprintarXiv licencearXiv

From Pythagorean Runs to Pell-Generated Cubic Runs

  • Anatoly Eydelzon
Open at sourcePDFDOICite

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

  1. Long runs of integers with small prime factors and the divisor function of n!
    Tristan Freiberg
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. Long runs and rare patterns of a random completely multiplicative function
    Brice Pouly
    PreprintCC BYFull text5 citedAlgebra & number theoryOpenAlex
  3. Initial Runs in Integer Compositions: Lambert Series and Generalized Divisor Sums
    Igor Kleiner
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. The density of long runs of non-r-free integers
    Xinyang Jiang
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. On concatenations of two k-generalized Pell numbers
    Cherif B. Deme, Kancou D. Fall, Khady Faye +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
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  -