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probability c+d square, find limit

Source: Putnam 1982 B3

October 1, 2021

probabilitynumber theorycombinatoricsmusic theory

Problem Statement

Let

p_n

be the probability that

c+d

is a perfect square when the integers

c

and

d

are selected independently at random from the set

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

. Show that

\lim_{n\to\infty}p_n\sqrt n

exists and express this limit in the form

r(\sqrt s-t)

, where

s

and

t

are integers and

r

is a rational number.

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