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$a_1=n^2-10n+23, a_1=n^2-9n+31, a_1=n^2-12n+46.$

Source: Albania JTST 2015

October 4, 2023
number theory

Problem Statement

For every positive integer nn{}, consider the numbers a1=n2−10n+23,a2=n2−9n+31,a3=n2−12n+46.a_1=n^2-10n+23, a_2=n^2-9n+31, a_3=n^2-12n+46. a) Prove that a1+a2+a3a_1+a_2+a_3 is even. b) Find all positive integers nn for which a1,a2a_1, a_2 and a3a_3 are primes.
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