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 $1le\; ile\; k$.

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 $prod\_\{xle\; k\}(n+x)$ has at least k distinct prime divisors.

ee also

*Prime gap

References

*mathworld|urlname=GrimmsConjecture|title=Grimm's Conjecture
*Guy, R. K. "Grimm's Conjecture." §B32 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 86, 1994.