Diophantine quadruples and quintuples modulo 4

Andrej Dujella
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 4, 1998, Number 4, Pages 160–164
Full paper (PDF, 2596 Kb)

Details

Authors and affiliations

Andrej Dujella
Department of Mathematics, University of Zagreb,
Bijenicka cesta 30, 10000 Zagreb, CROATIA

Abstract

A Diophantine m-tuple with the property D(n) is a set { a₁, a₂ , … a_m} of positive integers such that for 1 ≤ i < j ≤ m, the number a_ia_j + n is a perfect square. In the present paper we give necessary conditions that the elements a₁ of a set {a₁, a₂, a₃, a₄, a₅, a₆} must satisfy modulo 4 in order to be a Diophantine quintuple.

Keywords

• Diophantine m-tuple,
• P_n-set
• congruences

AMS Classification

• 11A07
• 11B75
• 11D79

References

1. J. Arkin, V. E. Hoggatt, E. G. Strauss, On Euler’s solution of a problem of Diophantus, Fibonacci Quart. 17(1979), 333-339.
2. A. Baker, H. Davenport, The equations 3×2 – 2 = y2 and 8×2 – 7 = z2, Quart. J. Math. Oxford Ser. (2) 20(1969), 129-137.
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, 1992, 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, On Diophantine quintuples, Acta Arith. 81(1997), 69-79.
8. H. Gupta, K. Singh, On k-triad sequences, Internat. J. Math. Math. Sci. 5(1985), 799-804.
9. S. P. Mohanty, A. M. S. Ramasamy, On Prtk sequences, Fibonacci Quart. 23(1985), 36-44.
10. V. K. Mootha, On the set of numbers {14,22,30,42,90}, Acta Arith. 71(1995), 259-263.
11. V. K. Mootha, G. Berzsenyi, Characterizations and extendibility of Pt-sets, Fibonacci Quart. 27(1989), 287-288.