Fermat-Like Equations that are not Partition Regular

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}.

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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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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).

    Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

  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.

    Google Scholar 

  11. [11]

    D. Gunderson, N. Hindman and H. Lefmann: Some partition theorems for infinite and finite matrices, Integers 14 (2014), Article A12.

  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).

    Google Scholar 

  13. [13]

    N. Hindman: Monochromatic Sums Equal to Products in ℕ, Integers 11A (2011), Article 10, 1–10.

    MathSciNet  Google Scholar 

  14. [14]

    N. Hindman, I. Leader and D. Strauss: Extensions of infinite partition regular systems, Electron. J. Combin. 22 (2015), Paper # P2.29.

  15. [15]

    P. A. Loeb and M. Wolff (eds.), Nonstandard Analysis for the Working Mathematician 2nd edition, Springer, 2015.

    Google Scholar 

  16. [16]

    L. L. Baglini: Partition regularity of nonlinear polynomials: a nonstandard approach, Integers 14 (2014), Article 30.

  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.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    M. Riggio: Partition Regularity of Nonlinear Diophantine Equations, Master Thesis, Università di Pisa, 2016.

    Google Scholar 

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Correspondence to Mauro Di Nasso.

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Di Nasso, M., Riggio, M. Fermat-Like Equations that are not Partition Regular. Combinatorica 38, 1067–1078 (2018). https://doi.org/10.1007/s00493-016-3640-2

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Mathematics Subject Classification (2000)

  • 03H05
  • 03E05
  • 05D10
  • 11D04