Mirela Jukić Bokun, Ivan Soldo: Triangular $D(-1)$-tuples, 233-239

Abstract:

In this paper, we consider the extendibility of the triangular $D(-1)$-tuples, i.e., the sets of the positive integers with the property that the product of any two of them decreased by 1 is the triangular number. We prove that the only triangular $D(-1)$-triples of the form $\{1, 2, c\}$, $c=2^np$, where $n$ is a non-negative integer and $p$ is a prime, are those with $c\in\{11,46,352,11936\}$. In addition, we prove that for these $c$'s no triangular $D(-1)$-quadruple of the form $\{1, 2, c, d\}$ exists.

Key Words: Diophantine $m$-tuple, Pellian equation, quadratic Diophantine equation, triangular number.

2020 Mathematics Subject Classification: 11D09, 11D59, 11Y50.

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