On the prime divisors of elements of a $D(-1)$ quadruple

Abstract

We show that if {1, b, c, d} is a D(-1) diophantine quadruple with b<c<d and c=1+s^2, then the cases s=p^k, s=2p^k, c=p and c=2p^k do not occur, where p is an odd prime and k is a positive integer. For the integer d=1+x^2, we show that it is not prime and that x is divisible by at least two distinct odd primes. Furthermore, we present several infinite families of integers b such that the D(-1) pair {1, b} cannot be extended to a D(-1) quadruple. For instance, we show that if r=5p where p is an odd prime, then the D(-1) pair {1, r^2+1} cannot be extended to a D(-1) quadruple.