On the exponential Diophantine equation x2+pmqn=2ypx^2+p^mq^n=2y^p

We study the exponential Diophantine equation $x^2+p^mq^n=2y^p$ in positive integers $x,y,m,n$, and odd primes $p$ and $q$ using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability of this equation.Comment: 10 pages. To appear in `New Zealand J. Math.

FAMILY OF ELLIPTIC CURVES E(p,q)‎: ‎y2=x2-p2x+q2

In this paper we show that for any two primes p and q greater than 5, theelliptic curve E(p,q) : y2 = x3 − p2x + q2 has rank at least 2. We will also provide twoindependent points on E(p,q). Then we will show that, conjecturally, the family {E(p,q)}contains an infinite subfamily of rank three elliptic curves

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics included modular forms, varieties over finite fields, rational and integral points on varieties, class groups, and integer factorization