4

Part of 2024 German National Olympiad

Problems(1)

When is 133...3 prime?

Source: Germany 2024, Problem 4

k>2

be a positive integer such that the

k

n_k=133\dots 3

, consisting of a digit

1

k-1

3

is prime. Show that

24 \mid k(k+2)

number theorynumber theory proposed