Vandita Patel

PERFECT POWERS THAT ARE SUMS OF CONSECUTIVE CUBES

Euler noted the relation 63 = 33 + 43 + 53 and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.

On powers that are sums of consecutive like powers

Let

On the Carlitz Rank of Permutation Polynomials Over Finite Fields: Recent Developments

k \ge 2

On perfect powers that are sums of two Fibonacci numbers

k≥2 be even, and let r be a non-zero integer. We show that for almost all

Perfect Powers that are Sums of Squares of an Arithmetic Progression.

d \ge 2

On the difference between permutation poynomials over finite fields

d≥2 (in the sense of natural density), the equation