A problem of Diophantus and Dickson's conjecture

Abstract

A Diophantine m-tuple with the property D(n), where n is an integer, is defined as a set of m positive integers with the property that the product of its any two distinct elements increased by n is a perfect square. It is known that if n is of the form 4k + 2, then there does not exist a Diophantine quadruple with the property D(n). The author has formerly proved that if n is not of the form 4k + 2 and n is not in {;-15, -12, -7, -4, -3, -1, 3, 5, 7, 8, 12, 13, 15, 20, 21, 24, 28, 32, 48, 60, 84}; ; , then there exist at least two distinct Diophantine quadruples with the property D(n). The main problem of this paper is to consider the set U of all integers n, not of the form 4k + 2, such that there exist at most two distinct Diophantine quadruples with the property D(n). One open question is whether the set U is finite or not. It can be proved that if n in U and |n| > 48, then n can be represented in one of the following forms: 4k + 3, 16k + 12, 8k + 5, 32k + 20. The main results of the this paper are: If n in U {;-9, -1, 3, 7, 11}; ; and n = 3 (mod 4), then the integers |n - 1|/2, |n - 9|/2 and |9n - 1|/2 are primes, and either |n| is prime or n is the product of twin primes. If n in U {;-27, -3, 5, 13, 21, 45}; and n = 5 (mod 8), then the integers |n|, |n - 1|/4, |n - 9|/4 and |9n - 1|/4 are primes.