mersenneforum.org - Report top-5000 primes here

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[QUOTE=Mini-Geek;375758]<snip>
I'd be more inclined to "trust" a number that's passed multiple PRP tests (especially multiple types) than a number that's passed a single deterministic primality test.
[/QUOTE]

Agreed completely.

[QUOTE=Batalov;375760][URL="http://mersenneforum.org/showthread.php?t=19142"]It had happened before[/URL], though. So it is a good thing to blow on cold water ([URL="http://translation.babylon.com/english/Once+burned+by+milk+you+will+blow+on+cold+water/"]Cf.[/URL]), just in case.

One thing that is implied in the above improbability arguments is that "let's say, a random bit flipped -- but the result is still random!" That's not what happens, at least in some implementations. If the whole interim residue is zeroed in (there are hardware failures that do that), then the rest of exponentiation will stay all zeroes. The end result is implementation dependent. P95 catches those zero-ins early, other programs - we don't know for sure (and who has time to read the source, right?).[/QUOTE]

Based on the strong wording on the top-5000 site, I can understand people's reluctance after finding a strong PRP to only wait until there is a "sure fire" proof before submitting a top-5000 prime but there are a couple of more points that I'd like to make:
1. If a bad PRP is found due to a computer glitch, isn't that same glitch just as likely to incorrectly "prove" the "PRP" as "prime"?
2. The example pointed out by Serge is not one that I am referring to. I'm referring to examples where a non-power-of-2 base PRP is found and has yet to be proven like was the case with Dimitry here and has been the case for me in the past. In the example shown by Serge, since it was base 2 I believe there was no PRP test run and the actually primality proof was incorrect due to a computer glitch. A second test on the same computer likely would have incorrectly found it prime again.

Here is what would have to happen for there to be any real risk to reporting a PRP (of the k*b^n+1 or k*b^n-1 form) early, that is it has not been proven yet:

1. A computer would have to have a glitch that incorrectly finds a PRP but its proof finds it composite. Very unlikely. (If it finds it prime, then the bad prime gets reported to top-5000 anyway and eventually rejected so it doesn't matter whether you waited for the primality test or not.)

2. A good computer correctly finds a PRP but the PRP turns out to be a Carmichael number and so the same computer correctly finds the PRP as composite. This composite would have to be top-5000 size at the time that it is incorrectly reported as prime to top-5000. Not likely in any of our lifetimes.*

That's why I feel safe in reporting top-5000 PRPs while my machine is still working on their proof.

If people want to be "sure fire" as is reasonably possible, after a top-5000 size PRP is found, a different computer should be used to prove it prime. Since many people do not have access to their machines for days at a time, this is sure to be a burden.

*If anyone can quote the chances of a top-5000 size 3-PRP being composite, I would like to see them. I have heard figures like 1 in < 10^20. That would have a less chance of happening then winning 2 major lotteries with back-to-back tickets. Clearly computer glitches are more frequent and I believe such glitches are just as likely to report an incorrect PRP as they are to "prove" that same "PRP" as "prime", in which case it doesn't matter if one reported it to the top-5000 site before or after the "proof".

Indeed, for another example, for all prime p, all of the 3[SUP]p[/SUP]+-3[SUP](p+1)/2[/SUP]+1 are PRPs. (3-PRPs, to be exact. In default mode, base 3 is used. Well, devil is in the details, for each p, + or - sign is determined by a simple rule, and for the expression with the opposite sign one has to divide that complement expression by 7... - and [U]both of these[/U] will be PRPs!)

For example:
3^6689+3^3345+1 is 3-PRP and (3^6689-3^3345+1)/7 is 3-PRP
3^6709-3^3355+1 is 3-PRP and (3^6709+3^3355+1)/7 is 3-PRP
...you can go as high as you want, they will be 3-PRPs

But only the few of them are actually (2-)PRPs. One [I]must[/I] use -b2 for the project of searching for these.
And then, all known of those true (2-) PRPs are (Eisenstein-Mersenne) primes [SPOILER]see my "Location" field[/SPOILER]. There's also a known test for these that is faster than the N+-1 test (N+-1 test is also a valid test for them)

[QUOTE]I've been ploughing the megadigit terrain for quite some time with various conjectures. Let's try to have the megaprimes coming in those triplets that Gary mentions ;-) [/QUOTE]

You certainly are overdue. Start your own triplet.

[QUOTE=unconnected;375706]Can't wait another day to complete N+1 test. So I submit it right now.
Added [URL="http://primes.utm.edu/primes/page.php?id=118016"]118016[/URL] : 653*10^1435026-1 (1435029 digits)[/QUOTE]

Congratulation :smile:

May I come up with a suggestion:

We create a CRUS MegaPrime thread, where all primes with at least 1 million digits, that is found by CRUS members, is listed. I know the moderators may not like such a list, however MegaPrimes is for CRUS still a rare thing and there is now 3 moderators, so the task of maintaining the thread should be one that can be accomplished without being too much of a burden :smile:

Any thoughts?

[URL]http://primes.utm.edu/primes/page.php?id=118060[/URL]

115*26^520277-1 is prime 736181 digits and 214 entry rank

Conjecture proven

Results emailed

[QUOTE=MyDogBuster;376347][URL]http://primes.utm.edu/primes/page.php?id=118060[/URL]

115*26^520277-1 is prime 736181 digits and 214 entry rank

Conjecture proven

Results emailed[/QUOTE]

Congrats on a huge proof!! CRUS is really on a roll. That's 3 huge proofs and 4 huge primes in the last 2 weeks.

:bow wave::groupwave::george::chalsall::anurag:

Move discussion about sieving on a GPU vs a CPU to a new thread [URL="http://www.mersenneforum.org/showthread.php?t=19449"]here[/URL].

[QUOTE=MyDogBuster;376347]115*26^520277-1 is prime 736181 digits and 214 entry rank

Conjecture proven[/QUOTE][QUOTE=gd_barnes;376597]Congrats on a huge proof!! CRUS is really on a roll. That's 3 huge proofs and 4 huge primes in the last 2 weeks.[/QUOTE]
Wow, I glossed over those two important words "conjecture proven" the first time a read this...a hearty belated congratulations! :bow: It's always heart-warming to see one of the original bases <32 fall - proofs in that range are so far and few these days, and quite an accomplishment when they do come.

I keep toying with the idea of picking up S9 and trying to knock out its last k...maybe after we've gotten R6 to 2M.

R1024 just became a 1-ker

[QUOTE][COLOR=red]The prime "37*2^6xxxxx1-1" is unusually large. Primes this size require must be submitted manually by e-mailing [U]Chris Caldwell[/U] both the prime and an explanation how it was proven prime.[/COLOR][/QUOTE]Haven't seen this warning before. :rolleyes:
And maybe will never see again.

It has more than 2 million digits.