Note on a paper "An Extension of a Theorem of Euler" by Hirata-Kohno et al

In this paper we extend a result of Hirata-Kohno, Laishram, Shorey and Tijdeman on the Diophantine equation $n(n+d)...(n+(k-1)d)=by^2,$ where $n,d,k\geq 2$ and $y$ are positive integers such that $\gcd(n,d)=1.

On the Diophantine equation x2+q2m=2ypx^2+q^{2m}=2y^p

In this paper we consider the Diophantine equation $x^2+q^{2m}=2y^p$ where $m,p,q,x,y$ are integer unknowns with $m>0,$ $p$ and $q$ are odd primes and $\gcd(x,y)=1.$ We prove that there are only finitely many solutions $(m,p,q,x,y)$ for which $y$ is not a sum of two consecutive squares. We also study the above equation with fixed $y$ and with fixed $q.

Composite Rational Functions and Arithmetic Progressions

In this paper we deal with composite rational functions having zeros and poles forming consecutive elements of an arithmetic progression. We also correct a result published earlier related to composite rational functions having a fixed number of zeros and poles

On products of disjoint blocks of arithmetic progressions and related equations

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erd\H{o}s and Graham.Comment: 10 page

Trinomials ax8+bx+c with Galois groups of order 1344

Bruin and Elkies ([7]) obtained the curve of genus 2 parametrizing trinomials ax8 + bx + c whose Galois group is contained in G1344 = (ℤ/2)3 ⋊ G168. They found some rational points of small height and computed the associated trinomials. They conjecture that the only ℚ-rational points of the hyperelliptic curve Y2 = 2X6 + 28X5 + 196X4 + 784X3 + 1715X2 + 2058X + 2401 are given by (X, Y ) = (0, ± 49), (-1, ± 38), (-3, ± 32), and (-7, ± 196). In this paper we prove that the above points are the only S-integral points with S={2,3,5,7,11,13,17,19}