Grimm's conjecture

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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.

Contents 1 Formal statement 2 Weaker version 3 See also 4 References

[edit] 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 .

[edit] 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 has at least k distinct prime divisors.

[edit] See also

• Prime gap

[edit] References

• Eric W. Weisstein, Grimm's Conjecture at MathWorld.
• Guy, R. K. "Grimm's Conjecture." §B32 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 86, 1994.