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Grimms conjecture

If $n + 1$, $n + 2$, …, $n + k$ are all composite numbers, then there are k distinct primes $p_ᵢ$ such that $p_ᵢ$ divides $n + i$ for $1 \leq i \leq k$.

For example, for the range $242$ to $250$, one can assign distinct primes as follows:

$242: 11$ , $243: 3$ , $244: 61$ , $245: 7$ , $246: 41$ , $247: 13$ , $248: 31$ , $249: 83$ , $250: 5$

According to Grimm's (1969) , it's already proven that there are finitely many exceptions to this conjecture. (Reference from the topic of Grimm's problem from David wells prime numbers).

Paul Erdos and J.L.Selfridge has already shown that even if we consider the weak version of Grimm's conjecture it would imply Legendre's conjecture . ( Reference - Some problems on the prime factors of the consecutive integers 2 ) , In his paper he made many other important equivalencies to the Grimms conjecture, which I see far beyond my ken right now, In elementary abstract please someone reduce the levels of his argument and make us understand some of the important implications of Grimm's conjecture .

Moreover if it's already proven that it has finitely many exceptions, then how could one proceed to generate proof of Grimm's conjecture ?! And till date has been there any significant progress on it ? And if possible list few other conjectures related to Grimm's conjecture, and mostly on composite numbers and prime numbers ,

Thanks a lot in advance :) Regards

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There's a misinterpretation here of Grimm's exact wording on "finite exceptions." Specifically, Grimm shows that for any $C>n^{n-1}$, $\{C+1,C+2,\cdots,C+n\}$ has distinct prime factors $p_1,\cdots,p_n$ such that $p_i|C+i$. So in a sense it has been proven true for large enough $C$ and "arbitrarily long sequences."

This leaves all other cases open for example $\{C+1,\cdots,C+m\}$ where $m<n$ and $m\notin \{r: C>r^{r-1}\}$, e.g. "finite exceptions."

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  • $\begingroup$ Is it just a possibility which they couldn't cover and it's open to assume that there might be exceptions, and if yes then finite ?! $\endgroup$ – Alphatrion Oct 1 '19 at 20:26
  • $\begingroup$ If your asking whether the conjecture has at most finitely many exceptions, then I'm not sure. The above implies that there could still be infinitely many exceptions, as we don't have a proof for them yet. Afterall there are infinitely many such pairs $(C,m)$ as above. $\endgroup$ – Alex R. Oct 1 '19 at 20:30
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    $\begingroup$ Is there a simple algorithm to find the primes or is it just existence ? For $C$ large enough I'd expect picking the largest possible prime factor of $C+j$, then the largest possible not already taken prime factor of $C+j-1$ .. may work $\endgroup$ – reuns Oct 1 '19 at 22:36
  • $\begingroup$ another restatement of Grimm's is that the gaps between primes never exceeds the number of primes less than the gap. $\endgroup$ – user645636 Oct 3 '19 at 12:30

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