On the size of Diophantine m-tuples

Abstract

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy+n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n| <= 400 then |S| <= 32, and if |n| > 400 then |S| < 267.81 log|n| (log log|n|)^2. The question whether there exists an absolute bound (independent on n) for |S| still remains open.

16 pages. See also http://www.math.hr/~duje/papers.html

References in corpus (2)

• A Generalised Stern-Brocot Tree from Regular Diophantine Quadruples
• Some Rational Diophantine Sextuples

Cited by in corpus (14)

• Bounds for the size of sets with the property D(n)
• Triples which are $D(n)$-sets for several $n$'s
• Diophantine quadruples with the properties $D(n_1)$ and $D(n_2)$
• A Pellian equation with primes and applications to D(-1)-quadruples
• A Polynomial Variant of Diophantine Triples in Linear Recurrences
• Explicit constructions of Diophantine tuples over finite fields
• Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples
• $F$-Diophantine sets over finite fields
• On the largest element in D(n)-quadruples
• Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters
• Improved upper bounds on Diophantine tuples with the property $D(n)$
• A polynomial variant of a problem of Diophantus and its consequences
• Bipartite Diophantine tuples and their applications
• The extension of the $D(-k)$-pair $\{k,k+1\}$ to a quadruple