At 11:59 AM 1999/03/05 GMT, "Brian J Beesley" [EMAIL PROTECTED] wrote:
(Ken wrote)
Right. The odds heavily favor both mismatched residues being nonzero.
A zero residue at the last iteration is what indicates primality.
Depends on what you mean. If a Mersenne number that has been
tested once really is prime, but the test "went wrong", then we
have a wrong result i.e. calling the number composite when it isn't.
I mean that nearly all mersenne numbers are nonprime even if the exponents
are prime, so of the numbers we bother to LLtest, when an error occurs,
the odds are 99%+ that it does not conceal an actual Mersenne prime.
And in the unusal case of a nonprime Mersenne number being erroneously
identified as prime, it would be multiply-checked and so the error caught,
rather than being announced as a prime.
This is why double-checking is so important, if we want to find *all*
the Mersenne primes for exponents up to a given limit.
I agree that double checking is important in an exhaustive search,
and that exhaustive search (rather than haphazard or opportunistic search)
is valuable.
There *might* be one or two lurking somewhere in the mass of
exponents which have only been tested once.
Also what causes the errors, bugs in the code?
What I've seen most often is that prime95 and its relatives provide
early warning of unreliable hardware, whether cpu, RAM module, or
motherboard.
Usually caused by overheating - failed CPU fan, poor ventilation or
excessively hot environment, overclocking, poor thermal contact
between processor substrate and heatsink ...
The sorts of things I've seen include, in systems with good cpu heat sinking,
multiple case fans, cpu fans properly installed and no overclocking:
mechanically ill-fitting case causing a system to be unreliable
unreliable single SIMM out of a set of 4
unreliable CPU chip
unreliable motherboard
whether these component-level problems were due to electrical stress
(power outages, spikes, brownouts, static), thermal stress (overheating
or too-frequent power on/off), or latent defects in manufacture is unknown.
Ken
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