MathDB

a_k is smallest integer > a_{k-1} for which a_k +a_{k-1} is perfect square

Source: Dutch IMO TST 2018 day 1 p3

August 30, 2019

number theorySumfloor functionPerfect Square

Problem Statement

Let

n \ge 0

be an integer. A sequence

a_0,a_1,a_2,...

of integers is defined as follows: we have

a_0 = n

and for

k \ge 1, a_k

is the smallest integer greater than

a_{k-1}

for which

a_k +a_{k-1}

is the square of an integer. Prove that there are exactly

\lfloor \sqrt{2n}\rfloor

positive integers that cannot be written in the form

a_k - a_{\ell}

with

k > \ell\ge 0