Grimm's conjecture

In mathematics, and in particular number theory, Grimm's conjecture (named after Karl 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 $\prod _{x\leq k}(n+x)$ has at least k distinct prime divisors.

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

Prime number conjectures

Hardy–Littlewood 1st 2nd Agoh–Giuga Andrica's Artin's Bateman–Horn Brocard's Bunyakovsky Chinese hypothesis Cramér's Dickson's Elliott–Halberstam Firoozbakht's Gilbreath's Grimm's Landau's problems Goldbach's weak Legendre's Twin prime Legendre's constant Lemoine's Mersenne Oppermann's Polignac's Pólya Redmond–Sun Schinzel's hypothesis H Waring's prime number

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Grimm's conjecture

Formal statement

Weaker version

See also

References