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On the finiteness of solutions for polynomial-factorial Diophantine equations

  • Wataru Takeda ORCID logo EMAIL logo
Published/Copyright: December 12, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 2

Abstract

We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that l! is represented by NA⁢(x), where NA is a norm form constructed from the field norm of a field extension K/𝐐. We also deal with the equation NA⁢(x)=l!S, where l!S is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of 𝐙 and for any polynomial P⁢(x)∈𝐙⁢[x] of degree 2 or more the equation P⁢(x)=l!S has only finitely many solutions (x,l). For some special infinite subsets S of 𝐙, we can show the finiteness of solutions for the equation P⁢(x)=l!S unconditionally.

Keywords: Diophantine equation; finiteness of solutions; Brocard–Ramanujan problem; norm form
MSC 2010: 11D09; 11D45; 11D72; 11D85

Communicated by Valentin Blomer

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 19J10705

Funding statement: This work was supported by Grant-in-Aid for JSPS Research Fellow (Grant Number: 19J10705).

Acknowledgements

The author deeply expresses their sincere gratitude to Professor M. Ram Murty and Professor Andrzej Dąbrowski for fruitful discussions. The author also deeply thanks Professor Kohji Matsumoto and Professor Masatoshi Suzuki for their precious advice.

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Received: 2020-06-01
Revised: 2020-11-08
Published Online: 2020-12-12
Published in Print: 2021-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
  3. Free cyclic group actions on highly-connected 2n-manifolds
  4. The distinction problems for Sp4 and SO3,3
  5. Affine cones over cubic surfaces are flexible in codimension one
  6. Permutations of zero-sumsets in a finite vector space
  7. On the finiteness of solutions for polynomial-factorial Diophantine equations
  8. Galois action on Fuchsian surface groups and their solenoids
  9. On a Lévy process pinned at random time
  10. Borsuk–Ulam theorem for filtered spaces
  11. A non-commutative differential module approach to Alexander modules
  12. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
  13. Quantum modularity of partial theta series with periodic coefficients
  14. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
  15. Zeros of GL2 𝐿-functions on the critical line
  16. Weighted boundedness of multilinear Calderón commutators
  17. Two characterizations of central BMO space via the commutators of Hardy operators
  18. Weyl 𝑛-algebras and the Swiss cheese operad
  19. Syzygies in equivariant cohomology in positive characteristic
  20. Epsilon factors of symplectic type characters in the wild case
https://doi.org/10.1515/forum-2020-0138
Keywords for this article
Diophantine equation; finiteness of solutions; Brocard–Ramanujan problem; norm form

Articles in the same Issue

  1. Frontmatter
  2. Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
  3. Free cyclic group actions on highly-connected 2n-manifolds
  4. The distinction problems for Sp4 and SO3,3
  5. Affine cones over cubic surfaces are flexible in codimension one
  6. Permutations of zero-sumsets in a finite vector space
  7. On the finiteness of solutions for polynomial-factorial Diophantine equations
  8. Galois action on Fuchsian surface groups and their solenoids
  9. On a Lévy process pinned at random time
  10. Borsuk–Ulam theorem for filtered spaces
  11. A non-commutative differential module approach to Alexander modules
  12. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
  13. Quantum modularity of partial theta series with periodic coefficients
  14. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
  15. Zeros of GL2 𝐿-functions on the critical line
  16. Weighted boundedness of multilinear Calderón commutators
  17. Two characterizations of central BMO space via the commutators of Hardy operators
  18. Weyl 𝑛-algebras and the Swiss cheese operad
  19. Syzygies in equivariant cohomology in positive characteristic
  20. Epsilon factors of symplectic type characters in the wild case
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