Glasnik Matematicki, Vol. 43, No.2 (2008), 265-291.

ON THE EXTENSIBILITY OF DIOPHANTINE TRIPLES {k-1, k+1, 4k} FOR GAUSSIAN INTEGERS

Zrinka Franušić

Abstract.   In this paper, we prove that if {k-1, k+1, 4k, d}, where k Z[i] \ {0, ± 1}, d Z[i], is a Diophantine quadruple in Z[i], i.e. if (k-1)d+1, (k+1)d+1, 4kd+1 are perfect squares in Z[i], then d = 16k³-4k.

2000 Mathematics Subject Classification.   11D09, 11R11.

Key words and phrases.   Diophantine quadruples, simultaneous Diophantine equations, linear recurrence sequences.

DOI: 10.3336/gm.43.2.04

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