MathDB

1

Part of 2022 Saudi Arabia JBMO TST

Problems(2)

n + k are composites for all nice numbers n, when n+1, 8n+1 both squares

Source: 2022 Saudi Arabia JBMO TST 1.1

11/3/2022
The positive n>3n > 3 called ‘nice’ if and only if n+1n +1 and 8n+18n + 1 are both perfect squares. How many positive integers k≤15k \le 15 such that 4n+k4n + k are composites for all nice numbers nn?
number theoryPerfect Squares
p^5 + p^3 + 2 = q^2 - q

Source: 2022 Saudi Arabia JBMO TST 2.1

11/3/2022
Find all pairs of positive prime numbers (p,q)(p, q) such that p5+p3+2=q2−q.p^5 + p^3 + 2 = q^2 - q.
number theoryDiophantine equation