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Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

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Divisibility in products of prime numbers

Let $p_i$ be the i-th prime number and $h_k:= p_{i+1}p_{i+2}\cdots p_{i+k}$. Is there for every $i\in \mathbb{N}$ a $k\in \mathbb{N}$ such that $h_k\equiv 1 \mod p_1p_2\cdots p_i$?