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Combinatorica

, Volume 38, Issue 5, pp 1067–1078 | Cite as

Fermat-Like Equations that are not Partition Regular

  • Mauro Di Nasso
  • Maria Riggio
Original Paper
  • 33 Downloads

Abstract

By means of elementary conditions on coefficients, we isolate a large class of Fermat-like Diophantine equations that are not partition regular, the simplest examples being xn + ym = zk with k ∉ {n, m}.

Mathematics Subject Classification (2000)

03H05 03E05 05D10 11D04 

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References

  1. [1]
    M. Bennett, I. Chen, S. R. Dahmen and S. Yazdan: Generalized Fermat equations: a miscellany, Int. J. Number Theory 11 (2015), 1–28.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    V. Bergelson: Ergodic Ramsey Theory - un update, in: “Ergodic Theory of ℤ d -actions”, London Math. Soc. Lecture Notes Series 228 (1996), 1–61.Google Scholar
  3. [3]
    V. Bergelson, H. Furstenberg and R. McCutcheon: IP-sets and polynomial recurrence, Ergodic Theory Dynam. Systems 16 (1996), 963–974.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    V. Bergelson, J. H. Johnson Jr. and J. Moreira: New polynomial and multidimensional extensions of classical partition results, arXiv:1501.02408 (2015).zbMATHGoogle Scholar
  5. [5]
    P. Csikvári, K. Gyarmati and A. Sárközy: Density and Ramsey type results on algebraic equations with restricted solution sets, Combinatorica 32 (2012), 425–449.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    B. Barber, N. Hindman, I. Leader and D. Strauss: Partition regularity without the columns property, Proc. Amer. Math. Soc. 143 (2015), 3387–3399.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Di Nasso: A taste of nonstandard methods in combinatorics of numbers, in: “Geometry, Structure and Randomness in Combinatorics” (J. Matousek, J. Nešetřil, M. Pellegrini, eds.), CRM Series, Scuola Normale Superiore, Pisa, 2015.Google Scholar
  8. [8]
    M. Di Nasso: Hypernatural numbers as ultrafilters, Chapter 11 in [15], 443–474.Google Scholar
  9. [9]
    N. Frantzikinakis and B. Host: Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc., to appear. (Published electronically: March 1, 2016.)Google Scholar
  10. [10]
    R. Goldblatt: Lectures on the Hyperreals — An Introduction to Nonstandard Analysis, Graduate Texts in Mathematics 188, Springer, New York, 1998.zbMATHGoogle Scholar
  11. [11]
    D. Gunderson, N. Hindman and H. Lefmann: Some partition theorems for infinite and finite matrices, Integers 14 (2014), Article A12.Google Scholar
  12. [12]
    M. J. H. Heule, O. Kullmann and V. W. Marek: Solving and verifying the boolean Pythagorean Triples problem via Cube-and-Conquer, arXiv: 1605.00723 (2016).CrossRefzbMATHGoogle Scholar
  13. [13]
    N. Hindman: Monochromatic Sums Equal to Products in ℕ, Integers 11A (2011), Article 10, 1–10.MathSciNetGoogle Scholar
  14. [14]
    N. Hindman, I. Leader and D. Strauss: Extensions of infinite partition regular systems, Electron. J. Combin. 22 (2015), Paper # P2.29.Google Scholar
  15. [15]
    P. A. Loeb and M. Wolff (eds.), Nonstandard Analysis for the Working Mathematician 2nd edition, Springer, 2015.zbMATHGoogle Scholar
  16. [16]
    L. L. Baglini: Partition regularity of nonlinear polynomials: a nonstandard approach, Integers 14 (2014), Article 30.Google Scholar
  17. [17]
    J. Moreira: Monochromatic sums and products in ℕ, arXiv:1605.01469 (2016).Google Scholar
  18. [18]
    R. Rado: Studien zur Kombinatorik, Math. Z. 36 (1933), 242–280.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    M. Riggio: Partition Regularity of Nonlinear Diophantine Equations, Master Thesis, Università di Pisa, 2016.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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