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Acta Mathematica Hungarica

, Volume 159, Issue 2, pp 589–602 | Cite as

Diophantine \(\mathcal{S} \)-quadruples with two primes which are twin

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Abstract

We show that there are only finitely many pairs of twin primes \((p, p+2) \) such that there exists an\(\mathcal{S} \)-Diophantine quadruple in the sense of Szalay and Ziegler for the set\(\mathcal{S} \) of integers composed only of primes p and p + 2.

Key words and phrases

Diophantine equation linear form in logarithms 

Mathematics Subject Classification

11D61 

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Notes

Acknowledgements

The author thanks the referee for helpful comments and for detecting a numerical error in a previous version of the manuscript. This paper was written during the 2nd International Conference on Diophantine m-tuples and related topics held at Purdue University North-West in October, 2018. The author thanks the organisers for the invitation and support.

References

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    J. Esmonde and M. R. Murty, Problems in Algebraic Number Theory, 2nd ed., Graduate Texts in Mathematics, 190, Springer-Verlag (New York, 2005)Google Scholar
  2. 2.
    Laurent, M., Mignotte, M., Nesterenko, Yu.: Formes linéaires en deux logarithmes et déterminants d'interpolation. J. Number Theory 55, 285–321 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Szalay, L., Ziegler, V.: On an \(S\)-unit variant of Diophantine \(m\)-tuples. Publ. Math. Debrecen 83, 97–121 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    V. Ziegler, On the existence of \(S\)-Diophantine quadruples, arXiv:1807.02972 (2018)

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.School of MathematicsUniversity of the WitwatersrandWitwatersrandSouth Africa
  2. 2.King Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of MathematicsUniversity of OstravaOstrava 1Czech Republic

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