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


Anitha Srinivasan

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

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 pk or 2pk, 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.

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DOI: 10.3336/gm.50.2.01


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