Grimm's conjecture

In mathematics, specifically in number theory, Grimm's conjecture states that, given a set of consecutive composite numbers, each element of the set is divisible by a prime that does not divide any of the other elements. It was first proposed by Carl Albert Grimm in 1969.^[1]

Though still unproven, the conjecture has been verified for all ${\displaystyle n<1.9\times 10^{10}}$.^[2]

Formal statement

If ${\displaystyle n+1,n+2,\dots ,n+k}$ are all composite numbers, then, for each ${\displaystyle i}$ such that ${\displaystyle 1\leq i\leq k}$, there is a prime ${\displaystyle p_{i}}$ that divides ${\displaystyle n+i}$ and does not divide ${\displaystyle n+j}$ for ${\displaystyle 1\leq j\leq k}$; ${\displaystyle j\neq i}$.

Weaker version

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

${\displaystyle \prod _{1\,\leq \,x\,\leq \,k}(n+x)}$

has at least ${\displaystyle k}$ distinct prime divisors.^[3]

Consequences

If Grimm's conjecture is true, then

${\displaystyle p_{i+1}-p_{i}\ll {\Big (}{\frac {p_{i}}{\log p_{i}}}{\Big )}^{1/2}}$

for all consecutive primes ${\displaystyle p_{i}}$ and ${\displaystyle p_{i+1}}$.^[3] This goes well beyond what the Riemann hypothesis would imply about gaps between prime numbers: the Riemann hypothesis only implies an upper bound of ${\displaystyle O(p_{i}^{1/2}(\log p_{i}))}$.^[4]

See also

• Prime gap