Abstract
In this paper, given a pair of odd coprime integers δ and ɛ, we study the positive n such that (n 2 + 1)/2 has two divisors d 1 and d 2 summing up to δn + ɛ.
We prove an asymptotic formula for the average number of solutions to the Diophantine equation axy−x−y=n in which a is fixed and n varies.
Abstract
Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k 2, k 2+1, c, d} is a D(−k 2)-quadruple with c < d, then c = 1 and d = 4k 2+1. This extends the work of the first author [20] and that of Dujella [4].
Abstract
Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k 2, k 2 + 1, 4k 2 + 1, d} is a D(−k 2)-quadruple, then d = 1, and that if {k 2 − 1, k 2, 4k 2 − 1, d} is a D(k 2)-quadruple, then d = 8k 2(2k 2 − 1).
Abstract
Let A and k be positive integers. In this paper, we study the Diophantine quadruples
Abstract
Let A and k be positive integers. We study the Diophantine quadruples
Abstract
We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x 2 − dy 2 = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.
Abstract
We prove that if k is a positive integer and d is a positive integer such that the product of any two distinct elements of the set {k + 1, 4k, 9k + 3, d} increased by 1 is a perfect square, then d = 144k 3 + 192k 2 + 76k + 8.
