Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 42 min ago
PreprintCC BY 4.0arXiv

Pythagorean triples in level sets of completely multiplicative functions

  • Guilherme Azevedo
  • Joel Moreira
Open at sourcePDFDOICite

Abstract

We show that given completely multiplicative functions f1,…,fd taking values in the unit circle, there exist Pythagorean triples (i.e., integer solutions to x2+y2=z2) with fi(x),fi(y),fi(z) all arbitrarily close to $1$ for all i. This is a new special case of the conjecture that any finite colouring of ℕ has a monochromatic Pythagorean triple. Our proof combines vanishing averages for aperiodic functions with concentration estimates for pretentious functions. A similar proof is applied to obtain the analogous statement for more general equations of the form ax2+by2=cz2 whenever a,b,c are perfect squares satisfying the Rado's condition.

Related

  1. A new proof of two-color partition regularity of Pythagorean triples
    James Leng
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  2. An elementary proof of the 2-regularity of Pythagorean triples
    William J. Wesley
    arXivPreprintCC BY-NC-ND 4.0Full textCombinatorics & discrete mathematicsarXiv
  3. Long runs and rare patterns of a random completely multiplicative function
    Brice Pouly
    PreprintCC BYFull text5 citedAlgebra & number theoryOpenAlex
  4. A new family of deformed trigonometric functions and their corresponding Pythagorean-like relations: application to the nonlinear harmonic oscillator
    P. R. Gordoa, A. Pickering, D. Puertas-Centeno +1
    arXivPreprintarXiv licenceFull textGeneral mathematicsarXiv
  5. Refined Bohr Inequalities and Bohr Radii Involving Schwarz Mappings for Holomorphic Functions in Banach Spaces and JB*-Triples
    Nabadwip Sarkar, Pradip Das
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{azevedo2026pythagorean,
  title = {{Pythagorean triples in level sets of completely multiplicative functions}},
  author = {Guilherme Azevedo and Joel Moreira},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.04903},
  eprint = {2607.04903},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.04903},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Pythagorean triples in level sets of completely multiplicative functions
AU  - Guilherme Azevedo
AU  - Joel Moreira
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2607.04903
UR  - https://arxiv.org/abs/2607.04903
AB  - We show that given completely multiplicative functions f_1,…,f_d taking values in the unit circle, there exist Pythagorean triples (i.e., integer solutions to x^2+y^2=z^2) with f_i(x),f_i(y),f_i(z) all arbitrarily close to $1$ for all i. This is a new special case of the conjecture that any finite colouring of ℕ has a monochromatic Pythagorean triple. Our proof combines vanishing averages for aperiodic functions with concentration estimates for pretentious functions. A similar proof is applied to obtain the analogous statement for more general equations of the form ax^2+by^2=cz^2 whenever a,b,c are perfect squares satisfying the Rado's condition.
ER  -