# A conjecture on consecutive odd composite numbers

Can you provide a proof or a counterexample for the claim given below?

Inspired by Grimm's conjecture I have formulated the following claim:

Let $$n_1,n_2,\dots,n_k$$ be a sequence of $$k$$ consecutive odd numbers which are all composite. Let $$\operatorname{gpf}(n_i)$$ be the greatest prime factor of $$n_i$$. Then, all $$\operatorname{gpf}(n_i)$$, $$1 \le i \le k$$ are mutually different.

Try it yourself.

• "be a greatest" --> "be the greatest". also, what's $k$? Commented Oct 30, 2018 at 10:40
• $3,5,7,9$. $3$ is the greatest prime factor of $3$ and $9$ Commented Oct 30, 2018 at 10:43
• @mathworker21 $3,5,7$ are not composite numbers... Commented Oct 30, 2018 at 10:53
• ah, my apologies. what is $k$ though? Commented Oct 30, 2018 at 10:54
• If your conjecture is true, then for any prime $p$, there exists a prime between $p(p-2)$ and $p^2$. (There gpfs are $p$) Showing it is likely to be as hard as proving that there exists a prime between $n$ and $n+k\sqrt{n}$ for some constant $k$, which is open problem as of now. (Currently it's proven only for $n+n^{0.525}$) Commented Nov 1, 2018 at 15:38