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3

Part of 2020 Federal Competition For Advanced Students, P2

Problems(1)

Interesting sequence

Source: 2020 Austrian National Competition for Advanced Students, Part 2 problem 3

7/17/2020
Let aa be a fixed positive integer and (en)(e_n) the sequence, which is defined by e0=1e_0=1 and en=a+∏k=0n−1ek e_n=a + \prod_{k=0}^{n-1} e_k for n≥1n \geq 1.
Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence.
(Theresia Eisenkölbl)
number theoryprime numbersSequenceAustria