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Original scientific paper
https://doi.org/10.3336/gm.50.2.01

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

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

Fulltext: english, pdf (120 KB) pages 261-268 downloads: 415* cite
APA 6th Edition
Srinivasan, A. (2015). D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES. Glasnik matematički, 50 (2), 261-268. https://doi.org/10.3336/gm.50.2.01
MLA 8th Edition
Srinivasan, Anitha. "D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES." Glasnik matematički, vol. 50, no. 2, 2015, pp. 261-268. https://doi.org/10.3336/gm.50.2.01. Accessed 9 Dec. 2021.
Chicago 17th Edition
Srinivasan, Anitha. "D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES." Glasnik matematički 50, no. 2 (2015): 261-268. https://doi.org/10.3336/gm.50.2.01
Harvard
Srinivasan, A. (2015). 'D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES', Glasnik matematički, 50(2), pp. 261-268. https://doi.org/10.3336/gm.50.2.01
Vancouver
Srinivasan A. D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES. Glasnik matematički [Internet]. 2015 [cited 2021 December 09];50(2):261-268. https://doi.org/10.3336/gm.50.2.01
IEEE
A. Srinivasan, "D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES", Glasnik matematički, vol.50, no. 2, pp. 261-268, 2015. [Online]. https://doi.org/10.3336/gm.50.2.01

Abstracts
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.

Keywords
Diophantine m-tuples; binary quadratic forms; quadratic diophantine equation

Hrčak ID: 150127

URI
https://hrcak.srce.hr/150127

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