We prove that for any positive integers x, d and k with gcd (x, d)= 1 and 3< k< 35, the product
x (x+ d)⋯(x+ (k− 1) d) cannot be a perfect power. This yields a considerable extension of …

AN EXTENSION OF A THEOREM OF EULER 1. Introduction The theorem of Euler ([Eul80], cf.
[Mor69, p.21-22], [MS03]) referred in the Page 1 AN EXTENSION OF A THEOREM OF EULER …

If k is a sufficiently large positive integer, we show that the Diophantine equation
n(n+d)⋯(n+(k-1)d)=y^ℓ has at most finitely many solutions in positive integers n, d, y and ℓ …

We consider some variations on the classical method of Runge for effectively determining
integral points on certain curves. We first prove a version of Runge's theorem valid for higher …

Euler proved that the product of four positive integers in arithmetic progression is not a
square. Győry, using a result of Darmon and Merel, showed that the product of three coprime …

In this paper we consider Diophantine equations of the form f (x)= g (y) f(x)=g(y) where ff has
simple rational roots and gg has rational coefficients. We give strict conditions for the cases …

Power values of sums of products of consecutive integers Page 1 isid/ms/2015/15 October 12,
2015 http://www.isid.ac.in/∼statmath/index.php?module=Preprint Power values of sums of …

This article centres around the contributions of the author and therefore, it is confined to
topics where the author has worked. Between these topics there are connections and we …

Let k≥ 0, a≥ 1 and b≥ 0 be integers. We define the arithmetic function gk, a, b for any
positive integer n byIf we let a= 1 and b= 0, then gk, a, b becomes the arithmetic function that …

Let a, b, c, d be given nonnegative integers with a, d⩾ 1. Using Chebyshevʼs inequalities
for the function π (x) and some results concerning arithmetic progressions of prime numbers …