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No perfect squares in A-A

Source: 2022 China TST, Test 3 P5

April 30, 2022

number theoryPerfect Squares

Problem Statement

Show that there exist constants

c

and

\alpha > \frac{1}{2}

, such that for any positive integer

n

, there is a subset

A

of

\{1,2,\ldots,n\}

with cardinality

|A| \ge c \cdot n^\alpha

, and for any

x,y \in A

with

x \neq y

, the difference

x-y

is not a perfect square.

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