6

Part of 2023 China Northern MO

Problems(1)

infinite m = n^2+ n + 1 with pefect square factors

Source: China Northern MO 2023 p6 CNMO

5/5/2024

A positive integer

m

is called a beautiful integer if that there exists a positive integer

n

such that

m = n^2+ n + 1

. Prove that there are infinitely many beautiful integers with square factors, and the square factors of different beautiful integers are relative prime.

number theoryPerfect SquaresPerfect Square