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

2013/09/17 by Anitha Srinivasan, Srinivasan, Anitha
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1309.4347

arxiv created 2013/09/17 · arxiv updated 2013/09/18

Abstract

We show that if 1, b, c, d is a D(-1) diophantine quadruple with b<c<d and c=1+s², 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+x², 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, r²+1 cannot be extended to a D(-1) quadruple.

Related

• On the existence of S-Diophantine quadruples
• A note on the number of S-Diophantine quadruples
• The extension of the D(-k)-pair \k,k+1\ to a quadruple
• Finding all S-Diophantine quadruples for a fixed set of primes S
• S-Diophantine quadruples with \mathbfS=\2,q\