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An Infinite Two-Parameter Family of Diophantine Triples


In this paper, we study sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. It is shown that if the two smallest elements have the form \({ KA}^2\), \(4{ KA}^4 \pm 4 A\) for some positive integers A and K, and the third one is chosen canonically, then any such set consisting of three elements can be contained in a unique such set with four elements.

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  1. Arkin, J., Hoggatt, V.E., Strauss, E.G.: On Euler’s solution of a problem of Diophantus. Fibonacci Q. 17, 333–339 (1979)

    MathSciNet  MATH  Google Scholar 

  2. Bliznac Trebješanin, M., Filipin, A.: Nonexistence of \(D(4)\)-quintuples. J. Number Theory 194, 170–217 (2019)

  3. Bravo, J.J., Gomez, C.A., Luca, F.: Powers of two as sums of two \(k\)-Fibonacci numbers. Miskolc Math. J. 17, 85–100 (2016)

    Article  MathSciNet  Google Scholar 

  4. Bugeaud, Y., Dujella, A., Mignotte, M.: On the family of Diophantine triples \(\{k-1, k+1,16k^3-4k\}\). Glasgow Math. J. 49, 333–344 (2007)

    Article  MathSciNet  Google Scholar 

  5. Cipu, M.: Further remarks on Diophantine quintuples. Acta Arith. 168, 201–219 (2015)

    Article  MathSciNet  Google Scholar 

  6. Cipu, M., Filipin, A., Fujita, Y.: Bounds for Diophantine quintuples II. Publ. Math. Debrecen 88, 59–78 (2016)

    Article  MathSciNet  Google Scholar 

  7. Cipu, M., Fujita, Y., Mignotte, M.: Two-parameter families of uniquely extendable Diophantine triples. Sci. China Math. 61, 421–438 (2018)

    Article  MathSciNet  Google Scholar 

  8. Cipu, M., Fujita, Y., Miyazaki, T.: On the number of extensions of a Diophantine triple. Int. J. Number Theory 14, 899–917 (2018)

    Article  MathSciNet  Google Scholar 

  9. Dujella, A.: An absolute bound for the size of Diophantine \(m\)-tuples. J. Number Theory 89, 126–150 (2001)

    Article  MathSciNet  Google Scholar 

  10. Dujella, A.: There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566, 183–224 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Dujella, A., Pethő, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math. Oxf. Ser. (2) 49, 291–306 (1998)

    Article  MathSciNet  Google Scholar 

  12. Filipin, A., Fujita, Y., Togbé, A.: The extendibility of Diophantine pairs I: the general case. Glas. Mat. Ser. III(49), 25–36 (2014)

    Article  MathSciNet  Google Scholar 

  13. Filipin, A., Fujita, Y., Togbé, A.: The extendibility of Diophantine pairs II: examples. J. Number Theory 145, 604–631 (2014)

    Article  MathSciNet  Google Scholar 

  14. Fujita, Y.: The extensibility of Diophantine pairs \(\{k-1, k+1\}\). J. Number Theory 128, 322–353 (2009)

    Article  MathSciNet  Google Scholar 

  15. Fujita, Y., Miyazaki, T.: The regularity of Diophantine quadruples. Trans. Am. Math. Soc. 370, 3803–3831 (2018)

    Article  MathSciNet  Google Scholar 

  16. Gibbs, P.E.: Comput. Bull. 17, 16 (1978)

  17. He, B., Pu, K., Shen, R., Togbé, A.: A note on the regularity of the Diophantine pair \(\{k,4k\pm 4\}\). J. Théor. Nombres Bordeaux (to appear)

  18. He, B., Togbé, A.: On a family of Diophantine triples \(\{k, A^2k+2A, (A+1)^2k+2(A+1)\}\) with two parameters II. Period. Math. Hungar. 64, 1–10 (2012)

    Article  MathSciNet  Google Scholar 

  19. He, B., Togbé, A., Ziegler, V.: There is no Diophantine quintuple. Trans. Am. Math. Soc. (to appear)

  20. Laurent, M.: Linear forms in two logarithms and interpolation determinants II. Acta Arith. 133, 325–348 (2008)

    Article  MathSciNet  Google Scholar 

  21. The PARI Group, PARI/GP, version \(\mathtt{2\mathit{}\mathit{.\mathtt{3}}}.\mathtt{5}\), Bordeaux, 2010. Accessed 27 July 2010.

  22. Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, IL (2010)

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The authors thank the referee for his/her careful reading and helpful comments. The second author is supported by Croatian Science Fund Grant Number 6422. The third author is supported by JSPS KAKENHI Grant Number 16K05079.

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Correspondence to Yasutsugu Fujita.

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Communicated by Emrah Kilic.

Dedicated to Professor Maurice Mignotte, for his influence on all of us.

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Cipu, M., Filipin, A. & Fujita, Y. An Infinite Two-Parameter Family of Diophantine Triples. Bull. Malays. Math. Sci. Soc. 43, 481–498 (2020).

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