Sunday, October 12
Climb to a prime
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.