- For those of you who don't like the wowee-gee-whiz side of
mathematical investigations, I suggest you by-pass this post.
I was reviewing a long list of Carmichael numbers for ones that had
relatively dense factors. That is, if C=prod(pi) is a the prime
factorization of a Carmichael number, then I wanted min(pi)/max(pi)
small. I choose arbitrarily to consider only a ratio at
approximately 10 or less. (Off the cuff, the smallest I remember was
around 4, but I am not done with the list.) The fascinating thing I
saw happening often was that, given min(pi)=p, max(pi)=q=10p-9.
Before you start hitting reply to razz me, consider first off that
often was about once every 5 sheets of printed numbers. Consider
also that I realize, given a min prime p and a max prime q, there are
infinitely many formulas that connect p to q, q=f(p). The reasons
why I saw 10p-9 was the (human) preference for base 10. For
instance, if the minimum prime is p=151 then 10p-9=1501 and it just
(kinda) sticks OUT because of the base 10 representation. (BTW 1501
isn't prime, I realize this.)
If I get any results relating to that (after I get done with what I
am doing now), say a linear form pmax=f(pmin) with high correlation,
or for relatively large subsets of the sample, I will post this.