Grimm's conjecture

In mathematics, and in particular number theory, Grimm's conjecture 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 distinct primes $p_{i}$ such that $p_{i}|(n+i)$ for $1\leq i\leq k$.

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 $\prod _{x\leq k}(n+x)$ 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, 2nd ed. New York: Springer-Verlag, p. 86, 1994.