Grimm's conjecture

In mathematics, and in particular number theory, Grimm's conjecture (named after Carl Albert Grimm) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.

Formal statement

Suppose n + 1, n + 2, …, n + k are all composite numbers, then there are k distinct primes p_i such that p_i divides n + i for 1 ≤ i ≤ k.

Weaker version

A weaker, though still unproven, version of this conjecture goes: If there is no prime in the interval $[n+1, n+k]$, then $\prod_{x\le k}(n+x)$ has at least k distinct prime divisors.

See also

• Prime gap

References

• Weisstein, Eric W., "Grimm's Conjecture", MathWorld.
• Guy, R. K. "Grimm's Conjecture." §B32 in Unsolved Problems in Number Theory, 3rd ed., Springer Science+Business Media, pp. 133–134, 2004. ISBN 0-387-20860-7