Glasnik Matematicki, Vol. 41, No.2 (2006), 205-216.

THE NON-EXTENSIBILITY OF D(4k)-TRIPLES {1, 4k(k-1), 4k2+1} WITH |k| PRIME

Yasutsugu Fujita

Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
e-mail: fyasut@yahoo.co.jp


Abstract.   For a nonzero integer n, a set of m distinct positive integers {a1, ... , am} is called a D(n)-m-tuple if aiaj+n is a perfect square for each i, j with 1 ≤ i < jm. Let k be an integer with |k| prime. Then we show that the D(4k)-triple {1, 4k(k-1), 4k2+1} cannot be extended to a D(4k)-quadruple.

2000 Mathematics Subject Classification.   11D09, 11D45.

Key words and phrases.   Simultaneous Pell equations, Diophantine tuples.


Full text (PDF) (free access)

DOI: 10.3336/gm.41.2.03


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