We study the family of elliptic curves y² = x(x-a²)(x-b²) parametrized by Pythagorean triples (a,b,c). We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over ℚ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil...

We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.

We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.

Given a binary recurrence ${u_n}_{n\ge 0}$, we consider the Diophantine equation $u_{n_1}^{x_1}\cdots u_{n_L}^{x_L}=1$ with nonnegative integer unknowns $n_1,...,n_L$, where $n_i\ne n_j$ for 1 ≤ i < j ≤ L, $max|x_i|:1\le i\le L\le K$, and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.