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P5

Part of 2024 5th Memorial "Aleksandar Blazhevski-Cane"

Problems(1)

Orders running around in circles

Source: 5th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane" - Senior - Problem 5

1/29/2024
For a given integer k≥1k \geq 1, find all kk-tuples of positive integers (n1,n2,...,nk)(n_1,n_2,...,n_k) with GCD(n1,n2,...,nk)=1\text{GCD}(n_1,n_2,...,n_k) = 1 and n2∣(n1+1)n1−1n_2|(n_1+1)^{n_1}-1, n3∣(n2+1)n2−1n_3|(n_2+1)^{n_2}-1, ... , n1∣(nk+1)nk−1n_1|(n_k+1)^{n_k}-1.
Proposed by Pavel Dimovski
number theory