Glasnik Matematicki, Vol. 50, No. 2 (2015), 261-268.

D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES

Anitha Srinivasan

Abstract.   A D(-1)-quadruple is a set of positive integers {a, b, c, d}, with a < b < c < d , such that the product of any two elements from this set is of the form 1+n² for some integer n. Dujella and Fuchs showed that any such D(-1)-quadruple satisfies a=1. The D(-1) conjecture states that there is no D(-1)-quadruple. If b=1+r², c=1+s² and d=1+t², then it is known that r, s, t, b, c and d are not of the form p^k or 2p^k, where p is an odd prime and k is a positive integer. In the case of two primes, we prove that if r=pq and v and w are integers such that p²v-q²w=1, then 4vw-1>r. A particular instance yields the result that if r=p(p+2) is a product of twin primes, where p ≡ 1 (mod 4), then the D(-1)-pair {1, 1+r²} cannot be extended to a D(-1)-quadruple. Dujella's conjecture states that there is at most one solution (x, y) in positive integers with y < k-1 to the diophantine equation x²-(1+k²)y²=k². We show that the Dujella conjecture is true when k is a product of two odd primes. As a consequence it follows that if t is a product of two odd primes, then there is no D(-1)-quadruple {1, b, c, d} with d=1+t².

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

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

DOI: 10.3336/gm.50.2.01

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