MathDB

Sequece related to diophantine equation

Source: Balkan Mathematical Olympiad 2008 Problem 4

May 6, 2008

inequalitiesalgebrapolynomialmodular arithmeticinductionnumber theoryrelatively prime

Problem Statement

Let

c

be a positive integer. The sequence

a_1,a_2,\ldots

is defined as follows a_1\equal{}c, a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3 for all positive integers

n

. Find all

c

so that there are integers

k\ge1

and

m\ge2

so that a_k^2\plus{}c^3 is the

m

th power of some integer.