MathDB

We say that a positive integer

n

is fantastic if there exist positive rational numbers

a

and

b

such that

n = a + \frac 1a + b + \frac 1b.

(a) Prove that there exist infinitely many prime numbers

p

such that no multiple of

p

is fantastic. (b) Prove that there exist infinitely many prime numbers

p

such that some multiple of

p

is fantastic.
Proposed by Walther Janous, Austria

Problem Statement