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Certain amount of numbers t=x^3+y^2 in a given set

Source: VJIMC 2017, Category II, Problem 4

April 2, 2017

number theory

Problem Statement

A positive integer

t

is called a Jane's integer if

t = x^3+y^2

for some positive integers

x

and

y

. Prove that for every integer

n \ge 2

there exist infinitely many positive integers

m

such that the set of

n^2

consecutive integers

\{m+1,m+2,\dotsc,m+n^2\}

contains exactly

n + 1

Jane's integers.

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