In an introductory post on Grimm Machines, I give a narrative to suggest why the following link between a given algorithm S and Grimm's conjecture should be studied. In this post, I give a summary (as suggested by Wlodzimierz Holsztynski) as well as new information and a new technical question.

Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08

In an introductory post on Grimm Machines, I give a narrative to suggest why the following link between a given algorithm S and Grimm's conjecture should be studied. In this post, I give a summary (as suggested by Wlodzimierz Holsztynski) as well as new information and a new technical question.

In an introductory post on Grimm Machines, I give a narrative to suggest why the following link between a given algorithm S and Grimm's conjecture should be studied. In this post, I give a summary (as suggested by Wlodzimierz Holsztynski) as well as new information and a new technical question.

The literature shows connections to smooth numbers, prime gaps, number of factors of binomial coefficients and so on, and has a connection to another MathOverflow question Prime divisors of the respectively minimal binomial coefficients

In an introductory post on Grimm Machines, I give a narrative to suggest why the following link between a given algorithm S and Grimm's conjecture should be studied. In this post, I give a summary (as suggested by Wlodzimierz Holsztynski) as well as new information and a new technical question.

The literature shows connections to smooth numbers, prime gaps, number of factors of binomial coefficients and so on, and has a connection to another MathOverflow question Prime divisors of the respectively minimal binomial coefficients

In an introductory post on Grimm Machines, I give a narrative to suggest why the following link between a given algorithm S and Grimm's conjecture should be studied. In this post, I give a summary (as suggested by W.H.) as well as new information and a new technical question.

In an introductory post on Grimm Machines, I give a narrative to suggest why the following link between a given algorithm S and Grimm's conjecture should be studied. In this post, I give a summary (as suggested by Wlodzimierz Holsztynski) as well as new information and a new technical question.

Edit 2016.08.24 At W.H.'s polite request, I revised the title. I ran computations for $L$ out to $1.6*10^9$ and found about 100 additional intervals on which $L$ is not injective. I will try a modified version that combines S and $L$ and report back. It seems many (maybe all but one?) of the $L$ intervals have a Case I fix, because there are very few smooth numbers between consecutive primes, and powers of 2 seem to avoid most of the problematic intervals. Until another natural divisor map suggests itself to me as being a good candidate for a Grimm map, I am going with a modified version which runs S and $L$ and uses S unless $L$ works better.

I have a part of an idea which suggests why S works better. Every jump of a prime $p$ for a distance $kp$ requires that its target $n+kp$ have its largest prime factor at most $kp$. Thus small primes tend to skip over nonsmooth numbers in a fashion where I am trying to quantify nonsmooth. In any case, $L$ fails to be injective through multiples of many pairs of consecutive or nearby pairs of smooth numbers, while $S$ fails to be injective due to short jumps of a prime when longer jumps might normally be expected. END Edit 2016.08.24.