I took on my table of A195264 three years ago. I am

*still*chipping away at the currently-317 primary unknowns (out of 10000), extending ECM-factorization attempts from 2000 to 5000 curves. I'm not yet even half-way through the list. Occasionally

*factordb*will have found for me (over time) the large factors for up to 116-digit composites! And in a handful of cases I have evolved an unknown into a prime — which finishes

*those*unknowns from consideration, but there are plenty more.

I've been a little resentful that I hadn't picked the arguably aesthetically-better (one less arithmetic symbol with which to deal), historically better-known

*home prime*sequence (A037274) on which to while away my time. No matter now: John Conway has offered $1000 for a resolution (problem 5) to term #20 of A195264,

*not*term #49 of A037274 — which he could easily have picked instead. I don't believe that (mathematically) it makes any difference: one is as contrived as the other. Conway's point (I think) is that there exist easily-stated but impossible-to-prove conjectures about such sequences. To collect the $1000 for problem 5, you will almost certainly have to

*luck*into a prime in the evolution, because proving that it

*doesn't*evolve into a prime may well be impossible.

While the prize may spur some to further evolve A195265, it may also impede them from

*sharing*any results of that effort. I hope that's not the case.

Hey Hans! You're the only person a google search indicates ever worked on this problem. I found a number you might be interested in, but I'll be damned if I can find your email address... send me a note: lorentztrans@gmail.com

ReplyDeletehttps://www.youtube.com/watch?v=NT4UCRoQEQ4

ReplyDeleteHey Hans, I found a solution to Base 2!

(Also a solution to base 6, 8, 29, etc. via XB+Y=X^Y)

Hey Keith. Your base-two solution appears to be 1269 = 3*3*3*47. Of course how factorizations are expressed is somewhat arbitrary, but the way Conway's climb-to-a-prime was played was not. More specifically, repeating factors are expressed as powers. For example, 255987 = 3^3*19*499 which is, by the way, a "proper" base-two solution.

DeleteMy solution is actually a dual one which involves a 2-cycle. It does follow the rules and I checked it myself. However, how did you find that one-cycle?

DeleteAlso, my Python code shows it accounts for this?

DeleteBonus Clarification:

ReplyDelete(1269)1011110101=11^11*101111(3^3*47)

(1007)1111101111=10011*110101(19*53)

Got it. (Your 1269 binary has a zero missing.) I got my one-cycle from David Seal's post, here:

Deletehttp://list.seqfan.eu/pipermail/seqfan/2017-June/017685.html

David also gives a couple of two-cycles, including yours.

Speaking of which, I don't know David's contacts or whether he considered this, but it is possible to set up equations like

ReplyDeletex^c=x*b^2+c / x^(c-1)-c/x=b^2

for xyz(b) where c=yb+z.

I was wondering if you had any advice how to approach equations like this. I know you are a brute force kind of guy, but you may know a way to prove it is impossible or something like that.

You are right! I'm not much of a mathematician. Rather, I consider myself an empiricist.

Delete