On 12 May 2001, at 17:29, [EMAIL PROTECTED] wrote: > Ah, but if C is composite with no known factors, so is 2^C - 1. ... and 2^(2^C-1)-1, and 2^(2^(2^C-1)-1)-1, ... So there _isn't_ a "largest", unless you disallow this construct. providing, of course, that there is at least one such C. Actually there is an adequate and increasing supply of such C's - namely those values of 2^p-1 (for _prime_ p) which have been doublechecked by GIMPS/Primenet but which have no known factors. This situation is likely to persist unless someone can find a foolproof means of factorizing prime-exponent Mersenne numbers which executes at least as fast as the Lucas-Lehmer test. (Now there's a challenge!) > Or, to paraphrase the famous poem about the wee beasties: > > People brag about huge composites C, > And hope none larger will smite 'em, > But then 2^C - 1 is composite, too, > And so on, ad infinitum. Precisely - providing ... (see above) > > Hans Riesel made the same point in his reply of 26 Apr 2001, to which > Brian replied: > > > It is certainly true that, for composite c, 2^c-1 is composite. It > > does _not_ follow that no factors of 2^c-1 are known - even if c is > > itself a Mersenne number. > > OK, so what factors of 2^C - 1 are known, for C = 2^33219281-1 ? AFAIK, none. > Now it is possible that for C composite and f the smallest factor > of C (known or not), 2^C - 1 may in fact have a smaller factor than > 2^f - 1 (more on this below), but for C sufficiently large, it will > nearly always be more difficult to find a factor of 2^C - 1 than of C, > if for no other reason that arithmetic modulo C will be much cheaper > than arithmetic modulo the much larger 2^C - 1. Well - it's certainly feasible to look for factors of (for example) 2^(2^727-1)-1 using the mmfac program, & with luck a factor with a reasonably small k might be found. AFAIK not a great deal of effort has been expended. A great deal of effort _has_ gone into looking for a factor of 2^727-1 without success, & we're now pretty sure that the smallest factor is bigger than 10^50. The point is that, unless someone can demonstrate that there is a procedure for determining a factor of M(C) given a factor of M(M(C)), it is at least possible (maybe very unlikely, but _possible_) that there might be, now or at some future time, at least one number C such that a factor of M(M(C)) is known whilst there is no known factor of M(C). A rigorous demonstration of how to determine a factor of M(C) given any factor of M(M(C)) would be sufficient to demolish me, but meantime I stand by my statement. If someone could merely demonstrate that, if the smallest factor of c = 2^p-1 is 2kp+1 for some k, C = 2^c-1 and the smallest factor of C with the form 2Kc+1 is 2Kc+1 for some K, then K > k^x for some x > 0, this would show that searching for factors of C without knowledge of factors of c is probably _unlikely_ to be successful. But, even though this demonstration would be an advance in the mathematical sense, it would not prove the theoretical impossibility of finding a factor of C without knowing any factor of c. If the limit x could be raised to 1, then any systematic search for factors of C without knowledge of factors of c would be proved to be ridiculously wasteful of computing time, but again without proving that it couldn't possibly be successful, if tried. > > Please don't think that I'm picking on Brian Oh, I don't! It's important to be precise about claims like this! I'm reminded of the story of three scientists travelling by train from London to Edinburgh. As the train crosses the border into Scotland, they observe a black sheep in an otherwise empty field. Astrophysicist: "So, all Scottish sheep are black!" Physicist: "No, at least some Scottish sheep are black!" Mathematician (after long pause): "No, in Scotland there exists at least one sheep, at least one side of which appears to be black." Regards Brian Beesley 1775*2^332181+1 is prime! (100000 digits) Discovered 22-Apr-2001 _________________________________________________________________________ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
