On 14 Nov 2001, at 21:27, Nathan Russell wrote: > For better or for worse (I think for better - it should attract a few > searchers) the new Mersenne report has made it to the popular > technical/open-source/geek news site Slashdot.
Indeed. Downloads of Mersenne-related software from my anon ftp server are currently running at least 10x higher than normal. (So are attempts to misuse the server - it has been explicitly configured to reject "port bouncing" to a third party address - I'm not operating an anonymising service!) > [ ... snip ...] > -"The Primenet list has confirmed that while they still need to > totally test it out (which should be done by the 24th), they believe > that the number found today is the 39th positive." (To my knowledge, > only George has directly stated that he views the result to be > correct. While I don't question his word, even if he is correct, the > odds are in favor - but not *grossly* in favor - of the new prime > being the 39th Mersenne - NOT the 39th prime, which is similiar to an > error made in one of Cray's press releases IIRC) The chance of a false zero residual after precisely (n-2) iterations due to hardware glitches or "random" software errors are surely only 1 in (2^n-1). If a zero residual is _genuinely computed_ then surely the chances are very, very, very high that a Mersenne prime has been discovered. Conversely, and assuming that the probability of any single LL test being affected by a hardware glitch or data-value-independent software error is 0.01 (which is probably a low estimate), the probablity of two runs both going wrong independently and yet returning the same _64 bit_ residual is 1 in 10000 * 2^64. This is the approximate probability that a Mersenne prime will be missed _despite_ the double-checking procedure; it is of course very small, but is absolutely COLOSSAL compared with the probability of a false discovery _without_ verification. Verification is still, of course, neccessary in order to eliminate the possibility of a systematic error (in program or hardware); also, and probably most importantly, to eliminate absolutely the chance of a "forged" discovery. BTW there are a large number of small exponents (almost all under 1 million) for which there aren't even two matching 64 bit residuals - early programs often reported residuals to 16 bits, in some cases even less. For this reason I'm selectively triple checking these cases, using old systems which are too slow and/or too memory limited to be useful for anything else. I most certainly do _not_ expect to find any missed primes! If anyone wishes to assist in the task of cleaning up this anomaly in the database, please contact me privately. As to the chance of the newly discovered unverified number being _numerically_ the 39th Mersenne prime (as opposed to the order of discovery), surely the announcement that it is "probably" M39 has to wait until all smaller exponents have been LL tested once, and not confirmed until double-checking passes that exponent. Regards Brian Beesley _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
