Grimm's conjecture

Template:Short description In mathematics, specifically in number theory, Grimm's conjecture states that, for every set of consecutive composite numbers, there is an equally sized set of prime numbers, and a bijection that maps each composite in the former set to a prime in the latter set that it is divisible by. It was first proposed by Carl Albert Grimm in 1969.Template:Sfnp

Though still unproven, the conjecture has been verified for all $n < 1.9 \times 1 0^{10}$ .Template:Sfnp

Formal statement

If $n + 1, n + 2, \dots, n + k$ are all composite numbers, then there is a sequence of distinct prime numbers ${(p_{i})}_{i = 1}^{k}$ such that $p_{i}$ divides $n + i$ for $1 \leq i \leq k$ .

Weaker version

A weaker, though still unproven, version of this conjecture states that if there is no prime in the interval $[n + 1, n + k]$ , then

\prod_{1 \leq x \leq k} (n + x)

has at least $k$ distinct prime divisors.Template:Sfnp

Consequences

If Grimm's conjecture is true, then

p_{i + 1} - p_{i} ≪ (\frac{p_{i}}{\log p_{i}})^{1 / 2}

for all consecutive primes $p_{i}$ and $p_{i + 1}$ .Template:Sfnp This goes well beyond what the Riemann hypothesis would imply about gaps between prime numbers: the Riemann hypothesis only implies an upper bound of $O (\sqrt{p_{i}} (\log p_{i}))$ .Template:Sfnp