IIRC, the only known cases where 1+2^2^N is prime are N=0,1,2,3,4. Have you investigated any number theory properties of HF(N)? Any possible analog of Wilson's Theorem? (P is prime iff P divides 1+(P-1)! .) I have a vague recollection of something like this being true. The factorizations of N!+1 are suggestive of some more general rule, but I don't think there's an actual conjecture. Rich -------- Quoting Bill Gosper <billgosper@gmail.com>:

Is apparently the complete sequence of solutions to PrimeQ[1+Hyperfactorial@x]. (Checked through a couple of hundred.) (Unconvincing) probabilistic arguments "predict" finiteness of prime subsequences of (sufficiently) rapidly growing sequences, but {1, 2, 3} seems a little too pat. Maybe RCS can conjure one of his magic case arguments.

The complete list of prime 1+BarnesG seems to be 1, 2, 3, 4, 5, 9, 16. (Checked through 278.) ?rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun