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Conditional squares imply injectivity

Source: EMC 2023 Seniors P4

December 18, 2023

EMC 20232023functional equationfunctionIvan Novak orzevan orz

Problem Statement

Let

f\colon\mathbb{N}\rightarrow\mathbb{N}

be a function such that for all positive integers

x

and

y

, the number

f(x)+y

is a perfect square if and only if

x+f(y)

is a perfect square. Prove that

f

is injective.
Remark. A function

f\colon\mathbb{N}\rightarrow\mathbb{N}

is injective if for all pairs

(x,y)

of distinct positive integers,

f(x)\neq f(y)

holds.
Ivan Novak

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