engleski

The Extension of the D(-k)-pair {;k,k+1}; to a Quadruple

Let n be a non-zero integer. A set of m positive integers {; ; ; a_1, a_2, ... , a_m}; ; ; is called a D(n)- m-tuple if a_ia_j + n is a perfect square for all 1 <= i < j <= m. Let k be a positive integer. In this paper, we prove that if {; ; ; k, k+1, c, d}; ; ; is a D(-k)-quadruple with c>1, then d=1. The proof relies not only on standard methods in this field (Baker's linear forms in logarithms and hypergeometric method), but also on some less typical elementary arguments dealing with recurrences, as well as a relatively new method for determination of integral points on hyperelliptic curves.

Diophantine m-tuples ; Pellian equations ; hypergeometric method ; linear forms in logarithms

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