We show that if, we obtain the more precise conclusion that there are, in fact, no such
progressions. Our proofs exploit the modularity of Galois representations corresponding to …

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 …

arXiv:1112.2461v1 [math.NT] 12 Dec 2011 Page 1 BAKER’S EXPLICIT ABC-CONJECTURE
AND APPLICATIONS SHANTA LAISHRAM AND TN SHOREY Dedicated to Professor Andrzej …

In this paper, a number of results are deduced on the arithmetic structure of products of
integers in short intervals. By way of an example, work of Saradha and Hanrot, and of …

Let a, b, c be relatively prime positive integers such that a 2+ b 2= c 2. Jeśmanowicz'
conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine …

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 …

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 …

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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 …