MathDB

Fix a nonnegative integer

a_0

to define a sequence of integers

a_0,a_1,\ldots

by letting

a_k,k\geq 1

be the smallest integer (strictly) greater than

a_{k-1}

making

a_{k-1}+a_k{}

into a perfect square. Let

S{}

be the set of positive integers not expressible as the difference of two terms of the sequence

(a_k)_{k\geq 0}.

Prove that

S

is finite and determine its size in terms of

a_0.

Problem Statement