Problem 1056

Let $k\geq 2$ . Does there exist a prime $p$ and consecutive intervals $I_1,\ldots,I_k$ such that

\prod_{n\in I_i}n \equiv 1\pmod{p}

for all $1\leq i\leq k$ ?

Status

Open

No claim settles this problem.

Gu04: Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section A15 "Congruent products of consecutive numbers", printed p. 54, which records Erdős's example $3\cdot4\equiv5\cdot6\cdot7\equiv1\bmod11$ , asks for the least prime with three congruent products, and reports the least primes found for several numbers of products. Library home: Guy 2004.
Ma83: Mąkowski, Andrzej, On a number-theoretical problem of Erdős. Elem. Math. (1983), 101-102.