On Thursday 05 May 2005 20:13, you wrote:

> Finally, 7! doesn't produce any values. And thus Milton's conjecture is

> demolished, as about anything that he brainfarts on this list.

Moreover, a PARI/GP search plus a clever shell script using PFGW (guess I

should learn PFGW's scripting language instead, but...) confirms my previous

findings, and adds failures at n! for n = 10, 12-14, 16, 17, 22-64, 66-70,

72-135, 137-361, 363-683,685-900; that's as far as I'm willing to take this

search. I have reason to believe that only finitely many twin prime pairs are

generated by this process.

Additionally, in the course of running this code, I discovered a bug in

PARI/GP 2.2.9:

? isprime(42542905343533366778773944705953203289361426945380795183689)

*** isprime: impossible inverse modulo: Mod(0,

42542905343533366778773944705953203289361426945380795183689).

Ironically, Milton's work was not 100% in vain.

Décio

PS: in principle one can actually check my conjecture that the sequence is

finite: if p, p+2 is the largest twin prime pair generated, then test all n's

up to n = p. Obviously p! shares a factor with all twin primes in the set,

and so does n! for all n > p. Thus, no new twin prime pairs can be generated

from then on. Of course, exhausting the range up to n! when n is in the

thousands of digits isn't exactly practical.

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