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n^2 + n + 1 cannot be product of 2 integers with difference <2\sqrt{n}

Source: Dutch IMO TST3 2019 p2

January 11, 2020

Productnumber theory

Problem Statement

Let

n

be a positive integer. Prove that

n^2 + n + 1

cannot be written as the product of two positive integers of which the difference is smaller than

2\sqrt{n}