Glasnik Matematicki, Vol. 39, No.2 (2004), 199-205.

BOUNDS FOR THE SIZE OF SETS WITH THE PROPERTY D(n)

Andrej Dujella

Department of Mathematics, University of Zagreb, Bijenicka cesta 30, 10000 Zagreb, Croatia
e-mail: duje@math.hr

Abstract.   Let n be a nonzero integer and a1 < a2 < ... < am positive integers such that aiaj + n is a perfect square for all 1 ≤ i < j ≤ m. It is known that m ≤ 5 for n = 1. In this paper we prove that m ≤ 31 for |n| ≤ 400 and m < 15.476 log|n| for |n| > 400.

2000 Mathematics Subject Classification.   11D45, 11D09, 11N36.

Key words and phrases.   Diophantine m-tuples, property D(n), large sieve.

Full text (PDF) (free access)

DOI: 10.3336/gm.39.2.01

References:

  1. A. Baker and H. Davenport, The equations 3x2 - 2 = y2 and 8x2 - 7 = z2, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137.

  2. M. A. Bennett, On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math. 498 (1998), 173-199.

  3. E. Brown, Sets in which xy + k is always a square, Math. Comp. 45 (1985), 613-620.

  4. L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, New York, 1966, pp. 513-520.

  5. Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers, (I. G. Bashmakova, Ed.), Nauka, Moscow, 1974 (in Russian), pp. 103-104, 232.

  6. A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65 (1993), 15-27.

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

  8. A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Philos. Soc. 132 (2002), 23-33.

  9. A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.

  10. A. Dujella and A. Pethö, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), 291-306.

  11. P. X. Gallagher, A larger sieve, Acta Arith. 18 (1971), 77-81.

  12. P. Gibbs, Some rational Diophantine sextuples, preprint, math.NT/9902081.

  13. H. Gupta and K. Singh, On k-triad sequences, Internat. J. Math. Math. Sci. 5 (1985), 799-804.

  14. K. Gyarmati, On a problem of Diophantus, Acta Arith. 97 (2001), 53-65.

  15. S. P. Mohanty and A. M. S. Ramasamy, On Pr,k sequences, Fibonacci Quart. 23 (1985), 36-44.

  16. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94.

  17. A. Sárközy and C. L. Stewart, On prime factors of integers of the form ab + 1, Publ. Math. Debrecen 56 (2000), 559-573.

  18. I. M. Vinogradov, Elements of Number Theory, Nauka, Moscow, 1972 (in Russian).
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