Properties of balancing, cobalancing and generalized balancing numbers∗ Page 1 Annales
Mathematicae et Informaticae 37 (2010) pp. 125–138 http://ami.ektf.hu Properties of balancing …

A positive integer n is called a balancing number if 1+...+(n− 1)=(n+ 1)+···+(n+ r) for some
positive integer r. Balancing numbers and their generalizations have been investigated by …

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 …

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 …

We study rational points on the Erd\H {o} s-Selfridge curves\begin {align*} y^\ell= x (x+
1)\cdots (x+ k-1),\end {align*} where $ k,\ell\geq 2$ are integers. These curves contain" …

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

We prove that the product of k consecutive terms of a primitive arithmetic progression is
never a perfect fifth power when 3⩽ k⩽ 54. We also provide a more precise statement …

Let f (x, k, d)= x (x+ d) ⋯ (x+(k-1) d) f (x, k, d)= x (x+ d)⋯(x+(k-1) d) be a polynomial with k ≧ 2
k≥ 2, d ≧ 1 d≥ 1. We consider the Diophantine equation i= 1^ rf (x_i, k_i, d)= y^ 2∏ i= 1 rf …

We find all perfect $\ell $ th powers that may be expressed as a product of $ k-1$ terms from
a nontrivial coprime $ k $-term arithmetic progression, for $5\leq k\leq 8$ and $\ell\geq 3 …

ON A GENERALIZATION OF A PROBLEM OF ERDOS AND GRAHAM 1. introduction Let us
define f(x, k, d) = x(x + d) ···(x + (k − 1)d Page 1 ON A GENERALIZATION OF A PROBLEM OF …