Abstract The D (− 1)-quadruple conjecture states that there does not exist a set of four
positive integers such that the product of any two distinct elements is one greater than a …

Let A and k be positive integers. We study the Diophantine quadruples. We prove that if d is
a positive integer such that the product of any two distinct elements of the set increased by 1 …

In this paper, we show that some parametric families of D (− 1)-triples cannot be extended to
D (− 1)-quadruples. Using this result, we further show that in each case of r= pk, r= 2 pk, r 2+ …

Denote by Fn the nth Fibonacci number. We show that if a positive integer d satisfies the
property that for an integer k≥ 0 each of F2k+ 1d+ 1, F2k+ 3d+ 1 and F2k+ 5d+ 1 is a perfect …

On the D(−1)-triple {1,k2+1,k2+2k+2} and its unique D(1)-extension Page 1 Journal of Number
Theory 131 (2011) 120–137 Contents lists available at ScienceDirect Journal of Number …

ON THE FAMILY OF DIOPHANTINE TRIPLES {k + 1, 4k, 9k + 3} 1. Introduction Page 1
Periodica Mathematica Hungarica Vol. 58 (1), 2009, pp. 59–70 DOI: 10.1007/s10998-009-9059-6 …

In this paper, we study D (-1)-triples of the form {1, b, c} in the ring Z [-t], t> 0, for positive
integer b such that b is a prime, twice prime, and twice prime squared. We prove that in …

Abstract Quadruples (a,b,c,d) of positive integers a<b<c<d with the property that the product
of any two of them is one more than a perfect square are studied. Improved lower and upper …

THE EXTENSIBILITY OF D(−1)-TRIPLES {1, b, c} 1. Introduction Diophantus raised the
problem of finding four (positive rational) Page 1 THE EXTENSIBILITY OF D(−1)-TRIPLES {1 …

Let b= 2, 5, 10 or 17 and t> 0. We study the existence of D (− 1)-quadruples of the form {1, b,
c, d} in the ring. We prove that if {1, b, c} is a D (− 1)-triple in, then c is an integer. As a …