Advertisement

Periodica Mathematica Hungarica

, Volume 64, Issue 1, pp 1–10 | Cite as

On a family of diophantine triples {K,A 2 K + 2A, (A + 1)2 K + 2(A + 1)} with two parameters II

  • Bo He
  • Alain Togbé
Article

Abstract

Let A and k be positive integers. In this paper, we study the Diophantine quadruples
$$\{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1)d\} .$$
If d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then
$$\begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 + \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered} $$
for A ≥ 52330 and any k. This extends our result obtained in [4].

Key words and phrases

Diophantine tuples simultaneous Diophantine equations 

Mathematics subject classification numbers

11D09 11D25 11J86 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Dujella, An absolute bound for the size of Diophantine m-tuples, J. Number Theory, 89 (2001), 126–150.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math., 566 (2004), 183–214.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    B. He and A. Togbé, On the D(−1)-triple {1, k 2 + 1, k 2 + 2k + 2} and its unique D(1)-extension, J. Number Theory, 131 (2011), 120–137.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    B. He and A. Togbé, On a family of Diophantine triples {k,A 2 k + 2A, (A+1)2 k + 2(A + 1)} with two parameters, Acta Math. Hungar., 124 (2009), 99–113.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    M. Mignotte, Verification of a conjecture of E. Thomas, J. Number Theory, 44 (1993), 172–177.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    M. Mignotte, A corollary to a theorem of Laurent-Mignotte-Nesterenko, Acta Arith., 86 (1998), 101–111.MathSciNetMATHGoogle Scholar
  7. [7]
    T. Nagell, Introduction to Number Theory, Almqvist & Wiksell, Stockholm; Wiley, New York, 1951.MATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  1. 1.Department of MathematicsABa Teacher’s College WenchuanSichuanP. R. China
  2. 2.Department of MathematicsPurdue University North CentralWestvilleUSA

Personalised recommendations