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
Best Answer
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."