Original scientific paper

https://doi.org/10.3336/gm.49.2.03

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

Anitha Srinivasan ; Department of Mathematics, Saint Louis University-Madrid campus, Avenida del Valle 34, 28003 Madrid, Spain

Abstract

In [6] it was shown that if {1, b, c, d} is a D(-1) quadruple with b < c < d and b=1+r2, then r and b are not of the form r=pk, r=2pk, b=p or b=2pk, where p is an odd prime and k is a positive integer. We show that an identical result holds for c=1+s2, that is, the cases s=pk, s=2pk, c=p and c=2pk do not occur for the D(-1) quadruple given above. For the integer d=1+x2, 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, r2+1} cannot be extended to a D(-1) quadruple.

Keywords

Diophantine m tuples; binary quadratic forms; quadratic Diophantine equation

Hrčak ID:

130883

URI

https://hrcak.srce.hr/130883

Publication date:

18.12.2014.

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