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

Abstract

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In [6] it was shown that if {1, b, c, d} is a D(-1) quadruple with b < c < d and b = 1 + r(2), then r and b are not of the form r = p(k), r = 2p(k), b = p or b = 2p(k), where p is an odd prime and k is a positive integer. We show that an identical result holds for c = 1 + s(,)(2) that is, the cases s = p(k), s = 2p(k), c = p and c = 2p(k) do not occur for the D(-1) quadruple given above. For the integer d = 1 + x(2), we show that d 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.