Glasnik Matematicki, Vol. 49, No. 2 (2014), 275-285.

ON THE PRIME DIVISORS OF ELEMENTS OF A D(-1) QUADRUPLE

Anitha Srinivasan

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=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+s², 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+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.

2010 Mathematics Subject Classification.   11D09, 11R29, 11E16.

Key words and phrases.   Diophantine m tuples, binary quadratic forms, quadratic Diophantine equation.

DOI: 10.3336/gm.49.2.03

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