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Probability a random recursive sequence contains infinitely many primes

Source: SMMC 2024 A4

October 12, 2024
number theory

Problem Statement

Define a sequence by s0=1s_0 = 1 and for d≥1d \geq 1, sd=sd−1+Xds_d = s_{d-1} + X_d, where XdX_d is chosen uniformly at random from the set {1,2,…,d}\{1, 2, \dots, d\}.
What is the probability that the sequence s0,s1,s2,…s_0, s_1, s_2, \dots contains infinitely many primes?
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