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-   -   Lucky 13 (M51 related) (https://www.mersenneforum.org/showthread.php?t=23877)

 Prime95 2018-12-09 14:52

Lucky 13 (M51 related)

On Dec 7th, a new Mersenne prime was reported. I emailed the discoverer, obtained the last save file, and reran the last 100,000 iterations. Sure enough, we have a new prime!

Aaron has started a full test using prime95, but that doesn't count as an official double-check. We need a volunteer to run CudaLucas and/or mlucas.

As usual, the automatic email notification to myself and past Mersenne prime discoverers failed. Yes, the automatic email was debugged and tested after its last failure. Apparently bit rot set in. Fortunately the backup notification plan worked.

 Uncwilly 2018-12-09 15:13

:faf:

 ATH 2018-12-09 15:31

[QUOTE=Prime95;502046]There are other possible miracles[/QUOTE]

[url]https://mersenneforum.org/showpost.php?p=502046&postcount=2924[/url]

Nice hint :smile: It reminds me of the quote from Stephen King's "The Dark Tower" series:

"There are other worlds than these"

I am ready to test with CUDALucas, Mlucas etc.

 vasyannyasha 2018-12-09 15:44

Congratulations to discoverer!
Curtis Cooper is discoverer or not?

 Brian-E 2018-12-09 15:47

Quite astonishing!
How many more of these finds do we need before statistical cluster analysis lets us suspect that some as yet undiscovered mechanism is at work here making mersenne numbers more likely to be prime than was previously thought?

 pinhodecarlos 2018-12-09 16:10

Nice.

 ZK19 2018-12-09 16:48

Congratulations!

 ET_ 2018-12-09 18:01

DoubleMersennes.org (l.morelli at moreware.org) is silently waiting to know the exponent to start what could be considered a lengthy sieving process before updating the tables.

 TheJudger 2018-12-09 18:21

Congratulations!

 Batalov 2018-12-09 18:59

Congratulations!

I wonder if someone like Terence Tao could be intrigued into a review of the MP distribution heuristic, finally? :rolleyes:

 GP2 2018-12-09 19:26

I wonder if the exponent is congruent to 1 or to 3 (mod 4)?

There is a somewhat improbable imbalance in favor of the 1's.

 science_man_88 2018-12-09 19:39

[QUOTE=GP2;502193]I wonder if the exponent is congruent to 1 or to 3 (mod 4)?

There is a somewhat improbable imbalance in favor of the 1's.

even taking into account that all Sophie Germain primes that are 11 mod 12 are automatically not Mersenne prime exponents ?

 a1call 2018-12-09 19:45

[U]Perhaps[/U] we are dancing around a cluster which happens to contain (by chance) primes rather than Mersenne-Numbers with relatively large factors. If so we should expect things to even out further down the road.

 Prime95 2018-12-09 20:09

[QUOTE=ATH;502171]I am ready to test with CUDALucas, Mlucas etc.[/QUOTE]

I've sent you the exponent to begin official double-checking.

 Prime95 2018-12-09 20:23

[QUOTE=GP2;502193]I wonder if the exponent is congruent to 1 or to 3 (mod 4)?[/QUOTE]

1 mod 4

 ATH 2018-12-09 20:57

...and the exponent is 5 (mod 8).

 ewmayer 2018-12-09 20:57

This would of course have to happen just after George embarks on a 10-day preholiday ocean cruise ... George leaving town remains our best predictor of a new M-prime discovery. :)

[QUOTE=GP2;502193]I wonder if the exponent is congruent to 1 or to 3 (mod 4)?

There is a somewhat improbable imbalance in favor of the 1's.

I don't think it's giving too much away (well, cutting the potential exponents in half is actually giving a lot away) to say we're chalking up another for (exponent mod 4) = 1

 Batalov 2018-12-09 21:12

[QUOTE=Madpoo;502207]I don't think it's giving too much away (well, cutting the potential exponents in half is actually giving a lot away) to say we're chalking up another for (exponent mod 4) = 1[/QUOTE]
I must agree that this is indeed not giving away too much. :rolleyes:

[SPOILER]given posts number N-2 and N-3[/SPOILER]

 Dr Sardonicus 2018-12-09 21:47

Congratulations to the lucky discoverer (assuming it checks out), and to GIMPS! Good grief, that's less than a year since the last one!
[QUOTE=Prime95;502200]1 mod 4[/QUOTE]
Ooooh! That's 50 odd exponents, with 31 of them congruent to 1 (mod 4) and 19 of them congruent to 3 (mod 4). And 19 and 31 are :drama: [i]consecutive Mersenne prime exponents![/i]

I don't know what it all means, but it [i]must[/i] mean [i]something[/i]... :cmd:

 GP2 2018-12-09 22:25

[QUOTE=science_man_88;502194]even taking into account that all Sophie Germain primes that are 11 mod 12 are automatically not Mersenne prime exponents ?[/QUOTE]

There are just not that many Sophie Germain primes. The number of general primes goes as n/log n, whereas the number of Sophie Germain primes goes as n/(log n)^2

For example, [URL="https://en.wikipedia.org/wiki/Sophie_Germain_prime"]Wikipedia article[/URL] mentions that there are only 190 Sophie Germain primes less than 10[SUP]4[/SUP] and 56,032 Sophie Germain primes less than 10[SUP]7[/SUP], but there are 1,229 and 664,579 general primes respectively below those bounds.

So that can't account for the discrepancy.

 ixfd64 2018-12-09 22:46

Silly question: now that new Mersenne primes are no longer marked as a "success" until they are verified, does this mean that there is no way to spot them early unless George tells us?

[QUOTE=ixfd64;502223]Silly question: now that new Mersenne primes are no longer marked as a "success" until they are verified, does this mean that there is no way to spot them early unless George tells us?[/QUOTE]

The guessing game has a long history. In a way, it's half the fun. I think we've done a fair job of keeping details hidden on the server itself. In years past it was pretty obvious that a prime was found because the report page would show a success in a particular 1M range. LOL

The details of hiding the discovery until it was verified and ready to be announced has evolved to the point where nowadays most of the clues have come from people who know the exponent (because they're doing validation runs or whatever) and leak out enough info that guessing the exponent has become almost trivial.

I suppose the best clues for me would be the ones that require people to do a little mathy-stuff to figure it out, but are also general enough that it's not actually giving it away, but rather just narrows it down a bit. Like knowing whether exponent mod 4 is a 1 or a 3 ... narrows it down by half. LOL Or knowing that it's also exponent mod 8 = 5.

Of the known Mersenne primes, those 2 factoids account for 11 of the 50 (51 if we count the new one) ... 21.6%. For the entire set of prime exponent candidates from 2 through 1e9 there are 25% meeting that criteria so we're underperforming a bit there. :smile:

Of course we'll still get the people who complain that it takes too long between initial discovery and the press release going out, but whatever. :smile: These things take time...it has to be verified, George has to put together the PR stuff and use his contacts to get the word out, plus there is the timing of making the announcement when it's optimal and won't be buried in other news.

The aim of the PR is not only to announce it but also draw some attention to the project to get more participation, hopefully. After previous discoveries, there's a sharp uptick in people using the software although it tends to taper back down. Still, there are some who hear about it, think it's cool, and stick around at least long enough to do some solid tests.

With all the recent improvements in the code in just the past year, I think the extra attention would be great for people who may not be aware of those additions (PRP, the new error checking methods like Gerbicz and Jacobi, enhancements for newer CPUs and multithreading benchmarks, etc)

[QUOTE=Madpoo;502236]...Of the known Mersenne primes, those 2 factoids account for 11 of the 50 (51 if we count the new one) ... 21.6%. For the entire set of prime exponent candidates from 2 through 1e9 there are 25% meeting that criteria so we're underperforming a bit there...[/QUOTE]

I realized that if I narrowed it down even more and mentioned what the result is mod 18, and if you knew an approximate range (within 5M) of where the exponent is, and account for known factors and tests already reported in, you could narrow it down to a pool of 933 exponents, so that's a little too specific.

From there you might be able to narrow it down to assignments made within the past XX days (bearing in mind, some exponents are assigned well in advance and just take a while to even start, much less the time it takes to complete). But still, you might get it narrowed to a few hundred.

Anyway, so, you kind of see how the game is played. :smile:

 moebius 2018-12-10 04:31

Congratulations, hopefully it is also the largest currently known prime number:toot::toot::toot::toot:

 axn 2018-12-10 04:47

[QUOTE=Madpoo;502236]The guessing game has a long history.[/QUOTE]

That was not the point. The point is, nobody will look for a prime, because there are no more indications on the server that a prime was reported. You can't guess something, if you don't know that there is a "something" to guess.

[QUOTE=axn;502240]That was not the point. The point is, nobody will look for a prime, because there are no more indications on the server that a prime was reported. You can't guess something, if you don't know that there is a "something" to guess.[/QUOTE]

The reason it's hidden on the server from the start is because we've had a history of false positives. Most of them lately have been from a funny version of cudalucas 2.05.1 but there have been others. Couple years back a user was reporting quite a few of them from one of his machines, and earlier this year there was a false positive from a double-check in the 47-48M range or something like that.

By waiting until we've at least made sure it's not bogus, we cut down the noise. If people saw all of the false alarms it wouldn't be nearly as fun when the real thing happens. You're welcome. LOL

 axn 2018-12-10 05:00

Right. Which led to...
[QUOTE=ixfd64;502223]Silly question: now that new Mersenne primes are no longer marked as a "success" until they are verified, does this mean that there is no way to spot them early unless George tells us?[/QUOTE]

So, the answer to that is a "yes", I guess.

[QUOTE=axn;502242]So, the answer to that is a "yes", I guess.[/QUOTE]

Yeah, I suppose so. And now we also know the reason why it's done that way. False positives.

 a1call 2018-12-10 05:45

I found the detective work very satisfying last time.
Thank you for making math fun.
BTW Anyone looking for Twins this time around?
That could end up being a clue.:smile:

 henryzz 2018-12-10 09:04

Is it prime or prp currently? This could the first using the new testing methodology

 LaurV 2018-12-10 10:17

Sorry I lost the "start" of the thread. Congrats to the finder.

If the offer for DC still stands, I can put one card on it (say ~120 GHzD/D card).

(Edit: contrary to other "competitors" here, people know that I [U]always[/U] run CL in [U]two[/U] cards in parallel, with different shifts, so any mismatch or false residue is excluded).

(Edit 2: and yes, that is ~240 GHzD/D totally).

 axn 2018-12-10 13:13

What is the progress/ETA on the verification runs?

And, what is the plan on the press release? Pre-christmas or post new year?

 pinhodecarlos 2018-12-10 14:30

[QUOTE=axn;502269]What is the progress/ETA on the verification runs?

And, what is the plan on the press release? Pre-christmas or post new year?[/QUOTE]

I’ve verified on my TI-89, just awaiting for the slowest ones.

 GP2 2018-12-10 14:48

[QUOTE=henryzz;502254]Is it prime or prp currently? This could the first using the new testing methodology[/QUOTE]

It must have been found by LL test, because we know that Mlucas and CUDALucas are being used for the verification, not gpuOwL.

However, it might be worthwhile to retest every known Mersenne prime with PRP.

Currently our verification method is to use different software implementations of the same Lucas-Lehmer algorithm on different hardware (GPU vs CPU). PRP, on the other hand, is a different algorithm.

No, it can't prove primality. But it can prove compositeness. And at this point, with such an improbable surplus of Mersenne primes, maybe a drastic sanity check is needed for peace of mind, just in case software implementers exchanged ideas and somehow ended up all making some identical implementation error. It's a crazy notion, I know.

 kriesel 2018-12-10 15:07

[QUOTE=Prime95;502167]On Dec 7th, a new Mersenne prime was reported. I emailed the discoverer, obtained the last save file, and reran the last 100,000 iterations. Sure enough, we have a new prime!

Aaron has started a full test using prime95, but that doesn't count as an official double-check. We need a volunteer to run CudaLucas and/or mlucas.
[/QUOTE]
If it still matters, you have my standing offer from some time ago. Current fastest hardware is GTX1080Ti ready with CUDALucas v2.06 May 5 2017 beta, capable of knocking out a ~82M exponent in about 3 days, backed up by a GTX1080, and a 16-core e5-2670 system with prime95 V29.4b8. The GTX1080Ti at 84+GhzD/day for 85M is ~10% faster than a 2080 Ti for LL per [URL]https://www.mersenne.ca/cudalucas.php[/URL] (14th fastest model in the list, and that includes combined throughputs of some duals and a quad; 10th fastest single). And no, that 82M or 85M is not a clue to the nature of Mp51; it's just what I checked the 1080Ti's run time on recently. It would take me some time to put up mlucas on the system.

 kriesel 2018-12-10 15:19

[QUOTE=Prime95;502167]
As usual, the automatic email notification to myself and past Mersenne prime discoverers failed. Yes, the automatic email was debugged and tested after its last failure. Apparently bit rot set in.[/QUOTE]Tradition continues!

 Uncwilly 2018-12-10 15:26

I've noticed some interesting names with some interesting exponent assignments that will be "home for the holidays" as it were.

MD5 hash guess of one that looks tasty (includes exponent assignee MM/DD expected completion date)
[B]227a10ba5bbe1b4b033c2983ee417ee9[/B]

It does fit the 2 requirements posted by George and Alex. There also appears to be some funny business in the history on the assignment page.

The first verification run done by ATH confirms it: We have a new prime!

I'm not sure when George will do the press release - as was pointed out, the best way to predict when a prime is found is when George is on vacation. :smile:

I'll say a few things:[LIST][*]It was found by LL, not PRP. Although as noted, I think if we found one by PRP we'd do a pair of LL tests to confirm.[*]It's more than 80M and less than 90M[*]The discoverer has not found any previous primes[/LIST]

 irowiki 2018-12-10 16:14

So just to be sure, I set the lion's share of my machines to PRP. Should they be doing LL instead, or is PRP fine?

 Prime95 2018-12-10 16:23

[QUOTE=irowiki;502291]So just to be sure, I set the lion's share of my machines to PRP. Should they be doing LL instead, or is PRP fine?[/QUOTE]

PRP is a good way to go. I've moved half of my CPUs to PRP, eventually all will PRP.

[QUOTE=irowiki;502291]So just to be sure, I set the lion's share of my machines to PRP. Should they be doing LL instead, or is PRP fine?[/QUOTE]

Smarter folks than I can weigh in, like George did in mentioning that PRP is the future...

Here's my understanding: PRP is probabilistic and LL is deterministic. So although a PRP can say "M###" is [B]probably[/B] prime, an LL test can say "M###" [B]is[/B] prime.

PRP tests have the benefit of the Gerbicz error checking code that may make double-checking the tests unnecessary.

As to just how probable the PRP tests are, I understand that when PRP says it's probably prime, that probability is VERY high. And when I say the PRP error checking makes double-checks obsolete, I think that's generally true although I foresee cases where spot-checks make sense, but overall that's where I think it's headed. If I'm wrong in my assumptions, someone will correct me I'm sure. :smile:

When it comes to finding a new Mersenne Prime, I guess the question about whether a probable finding is good enough to be accepted, or would the LL test be required before it's confirmed. Maybe that's similar to how we already verify a prime using different software - it's a way of ensuring there aren't any code issues or other obscure things.

The [URL="https://www.mersenne.org"]mersenne.org[/URL] homepage has been updated with a little blurb about the discovery, linking to this thread. More of a pre-release release with a promise of details to follow.

 kriesel 2018-12-10 18:28

PRP and other runs on already-known mersenne primes

[QUOTE=GP2;502279]It must have been found by LL test, because we know that Mlucas and CUDALucas are being used for the verification, not gpuOwL.

However, it might be worthwhile to retest every known Mersenne prime with PRP.
[/QUOTE]
Looking back through mersenne.org, I see some exponents determined prime also have (anonymous) incorrect composite LL indications; [URL]https://www.mersenne.org/report_exponent/?exp_lo=216091&exp_hi=&full=1[/URL]
[URL]https://www.mersenne.org/report_exponent/?exp_lo=756839&exp_hi=&full=1[/URL]
[URL]https://www.mersenne.org/report_exponent/?exp_lo=6972593&exp_hi=&full=1[/URL]
[URL]https://www.mersenne.org/report_exponent/?exp_lo=13466917&exp_hi=&full=1[/URL]

Quite a few Mersenne primes also have recent failed TF or P-1 attempts also.

These following PRP runs on already known mersenne primes have not been reported yet to mersenne.org, but could be if it were useful. I may assemble them into my gpuowl reference data instead or also. Running known mersenne prime PRP checks was done as QA on gpuowl new versions/features.

gpuowl v5.0-df2bdf2 build for Win7 x64 PRP with P-1, p= 216091, 756839, 859433 by kriesel
[URL]https://www.mersenneforum.org/showpost.php?p=499569&postcount=867[/URL]
Per Preda, for a prime, residue = PRP-1 base, see
[URL]https://www.mersenneforum.org/showpost.php?p=499298&postcount=826[/URL]
[URL]https://www.mersenneforum.org/showpost.php?p=499292&postcount=823[/URL]

gpuowl v3.8-91c52fa p=1257787
[URL]https://www.mersenneforum.org/showpost.php?p=499299&postcount=827[/URL]

(p=756839, example of PRP GEC in [URL]https://www.mersenneforum.org/showpost.php?p=499252&postcount=815[/URL])

gpuowl v5.0-f604bb1 p=1257787, 1398269, 2976221, 3021377 by kriesel
[URL]https://www.mersenneforum.org/showpost.php?p=499263&postcount=818[/URL]

Also, unposted(?), gpuowl v5.0-9c13870 p=6972593

 Prime95 2018-12-10 18:41

[QUOTE=kriesel;502300]Looking back through mersenne.org, I see some exponents determined prime also have (anonymous) incorrect composite LL indications; [URL]https://www.mersenne.org/report_exponent/?exp_lo=216091&exp_hi=&full=1[/URL]
[URL]https://www.mersenne.org/report_exponent/?exp_lo=756839&exp_hi=&full=1[/URL]
[URL]https://www.mersenne.org/report_exponent/?exp_lo=6972593&exp_hi=&full=1[/URL]
[URL]https://www.mersenne.org/report_exponent/?exp_lo=13466917&exp_hi=&full=1[/URL][/QUOTE]

We ought to clean that junk out of the database. The server will reject duplicate prime results for known Mersenne primes. However, it seems the server is not rejecting composite results for known Mersenne primes.

 kriesel 2018-12-10 18:52

[QUOTE=Prime95;502301]We ought to clean that junk out of the database. The server will reject duplicate prime results for known Mersenne primes. However, it seems the server is not rejecting composite results for known Mersenne primes.[/QUOTE]
By all means clean out those erroneous composite results on known Mersenne primes. The TF and P-1 clutter on already-confirmed primes is a separate decision or two, which I feel should be removed, and prevented from reoccurring.

I'd like to see the false P-1 assignment expiration entries on unknown-status exponents gone too. (see [URL]https://www.mersenneforum.org/showthread.php?t=23817[/URL] for some known cases and a theory about a cause.)

[QUOTE=Uncwilly;502284]I've noticed some interesting names with some interesting exponent assignments that will be "home for the holidays" as it were.

MD5 hash guess of one that looks tasty (includes exponent assignee MM/DD expected completion date)
[B]227a10ba5bbe1b4b033c2983ee417ee9[/B]

It does fit the 2 requirements posted by George and Alex. There also appears to be some funny business in the history on the assignment page.[/QUOTE]

I'm trying to check your result but I can't quite figure out what your string looks like before hashing...

Do you mean it looks like "12345678Madpoo1210" or do you have spaces and the slash between MM/DD in there, or... ?

EDIT: wait, I tried adding spaces and the slash - I see the exponent you mean, and my hint that it's not a previous discoverer would rule that one out. :smile:

 kriesel 2018-12-10 19:34

[QUOTE=Madpoo;502297]Smarter folks than I can weigh in, like George did in mentioning that PRP is the future...

Here's my understanding: PRP is probabilistic and LL is deterministic. So although a PRP can say "M###" is [B]probably[/B] prime, an LL test can say "M###" [B]is[/B] prime.

PRP tests have the benefit of the Gerbicz error checking code that may make double-checking the tests unnecessary.

As to just how probable the PRP tests are, I understand that when PRP says it's probably prime, that probability is VERY high. And when I say the PRP error checking makes double-checks obsolete, I think that's generally true although I foresee cases where spot-checks make sense, but overall that's where I think it's headed. If I'm wrong in my assumptions, someone will correct me I'm sure. :smile:

When it comes to finding a new Mersenne Prime, I guess the question about whether a probable finding is good enough to be accepted, or would the LL test be required before it's confirmed. Maybe that's similar to how we already verify a prime using different software - it's a way of ensuring there aren't any code issues or other obscure things.[/QUOTE]
Very "on the money". And if there's a PRP discovery, I think it likely we confirm the PRP residue with a PRP run on different software & hardware, but the real mathematically acceptable proof will still be LL runs on different software and different hardware. In this case, does Gerbicz get included in the discoverers list, assuming it's prime95 or gpuowl that does the first PRP test, both of which implement the GEC?

 ewmayer 2018-12-10 20:04

[QUOTE=Madpoo;502297]When it comes to finding a new Mersenne Prime, I guess the question about whether a probable finding is good enough to be accepted, or would the LL test be required before it's confirmed. Maybe that's similar to how we already verify a prime using different software - it's a way of ensuring there aren't any code issues or other obscure things.[/QUOTE]
A PRP test will never suffice to claim a new prime discovery - that will always require a deterministic algorithm be run and independently verified.

[QUOTE=kriesel;502306]Very "on the money". And if there's a PRP discovery, I think it likely we confirm the PRP residue with a PRP run on different software & hardware, but the real mathematically acceptable proof will still be LL runs on different software and different hardware. In this case, does Gerbicz get included in the discoverers list, assuming it's prime95 or gpuowl that does the first PRP test, both of which implement the GEC?[/QUOTE]
Not to slight Robert G. in any way, but while the GEC is an important reliability improver for the programs using it, it is not a fundamental prime-finding algorithm, nor an actual software implementation thereof. It's an error-catching scheme, which someone actually needs to code it into an actual LL/PRP-test program. Put it this way: GIMPS doesn't give Crandall & Fagin credit in new-discovery name-lists - if anyone more should be added to such lists it should be those two, IMO. All of us who have written this type of code know there is a long list of folks upon whose ideas our work relies and any number of crucial software tools necessary for our work, but alas, you can't thank them all via discoverer credit.

 kriesel 2018-12-10 20:20

This would of course have to happen just after George embarks on a 10-day preholiday ocean cruise ... George leaving town remains our best predictor of a new M-prime discovery. :)[/QUOTE]
Maybe we should encourage George to travel more often.

 AG5BPilot 2018-12-10 20:34

Congratulations to the discover and everyone involved!

 kriesel 2018-12-10 20:35

[QUOTE=ewmayer;502308]A PRP test will never suffice to claim a new prime discovery - that will always require a deterministic algorithm be run and independently verified.

Not to slight Robert G. in any way, but while the GEC is an important reliability improver for the programs using it, it is not a fundamental prime-finding algorithm, nor an actual software implementation thereof. It's an error-catching scheme, which someone actually needs to code into an actual LL/[U]PRP[/U]-test program. Put it this way: GIMPS doesn't give Crandall & Fagin credit in new-discovery name-lists - if anyone more should be added to such lists it should be those two, IMO. All of us who have written this type of code know there is a long list of folks upon whose ideas our work relies and any number of crucial software tools necessary for our work, but alas, you can't thank them all via discoverer credit.[/QUOTE]
Point taken. The credits also don't explicitly list Lucas or Lehmer, or authors of the various factoring codes or algorithms that helped weed out so many other exponents early. I guess they could be considered lumped into the et al along with the thousands who ran composite primality tests and assorted factoring runs. Maybe candidates for a separate "Mersenne hunt hall of fame" listing though.

 preda 2018-12-10 21:05

Congrats! one MP every Christmas, they do look more frequent than expected, that's excellent news.

 chalsall 2018-12-10 21:21

[QUOTE=preda;502317]...they do look more frequent than expected, that's excellent news.[/QUOTE]

Or, perhaps put another way, that's interesting....

 kriesel 2018-12-10 21:30

[QUOTE=chalsall;502318]Or, perhaps put another way, that's interesting....[/QUOTE]
Or, not Eureka!, but Huh! That's funny... the more usual sound of impending discovery. Or it could be ominous, because if the current conjectures about frequency are close to accurate, we are now even more expectant of a relative drought, and in a much higher more computation-intensive range of exponents. I've been expecting for years, for the interval between discoveries to stretch out to 3 to 4 years. (So far I'm enjoying being wrong about that.) See [URL]https://www.mersenne.org/primes/[/URL]
Then again, we're only up to about 2010 on the double checking front.

 chalsall 2018-12-10 21:45

[QUOTE=kriesel;502320]Or, not Eureka!, but Huh! That's funny... the more usual sound of impending discovery. Or it could be ominous....[/QUOTE]

Yeah. Only time (and more work; many, many, years of work) will tell.... :smile:

 JeppeSN 2018-12-10 21:58

So GIMPS has found 17/51 or [B][I]one third[/I][/B] of the known Mersenne primes. That is something! I still cannot believe these numbers occur more frequently than conjectured. /JeppeSN

 Mark Rose 2018-12-10 22:17

And this time, I'm not aware of any leaks from the server. But then again, I haven't gone looking. :smile:

 ATH 2018-12-10 22:24

M( 8xxxxxxx )[B][SIZE="4"]P[/SIZE][/B], n = xxxxK, CUDALucas v2.06beta

Sorry about my absence just as my first verification run finished, but I just spent 8 hours in the emergency room with my dislocated shoulder (I'm ok now).

This was the Tesla V100 run and I have a MLucas run going and I had a another CUDALucas run going at a different FFT size on my Titan Black, but I might have to scrap it, as my computer is dead now that I came home, I think it is the power supply. What a great day :sad:

 ewmayer 2018-12-10 22:29

[QUOTE=ATH;502329]M( 8xxxxxxx )[B][SIZE="4"]P[/SIZE][/B], n = xxxxK, CUDALucas v2.06beta

Sorry about my absence just as my first verification run finished, but I just spent 8 hours in the emergency room with my dislocated shoulder (I'm ok now).[/QUOTE]

Ouch - is that a new injury or a recurring one? I have a "trick right shoulder" as result of a 35-years-ago skiing crash, every few years it again pops out ... luckily in my case I found a reliable repositioning that allows it to pop back in without needing anyone's assistance, after that it just needs lots of rest, ice and ibuprofen for about a week.

 chalsall 2018-12-10 22:52

[QUOTE=ewmayer;502331]...and ibuprofen for about a week.[/QUOTE]

Yeah. Be very careful what you do (and take) after an injury.

Sometimes your Doctors will be financially motivated to prescribe to you some seriously addictive stuff.

Cocaine nor Nicotine nor Caffeine have anything like the effects some doctors are regularly prescribing now-a-days.

 kriesel 2018-12-10 22:54

[QUOTE=ewmayer;502331]Ouch - is that a new injury or a recurring one? I have a "trick right shoulder" as result of a 35-years-ago skiing crash, every few years it again pops out ... luckily in my case I found a reliable repositioning that allows it to pop back in without needing anyone's assistance, after that it just needs lots of rest, ice and ibuprofen for about a week.[/QUOTE]
Pushing the limits in other ways may be a prerequisite here. Rotator cuffs, several major-joint sprains, a small break, various scars here.

 JeppeSN 2018-12-10 23:09

[QUOTE=chalsall;502326]Why? The empirical feeds into the theory; and tends to be taken more seriously in the analysis.

Belief is an artefact of presuppositions.[/QUOTE]

Has anyone come up with a "nicer" guess than [URL="https://primes.utm.edu/mersenne/heuristic.html"]https://primes.utm.edu/mersenne/heuristic.html[/URL]? /JeppeSN

 JeppeSN 2018-12-10 23:12

[QUOTE=Madpoo;502237]I realized that if I narrowed it down even more and mentioned what the result is mod 18, and if you knew an approximate range (within 5M) of where the exponent is, and account for known factors and tests already reported in, you could narrow it down to a pool of 933 exponents, so that's a little too specific.[/QUOTE] You could reveal if the exponent is 1 or 2 modulo 3. /JeppeSN

 Uncwilly 2018-12-10 23:35

[QUOTE=Mark Rose;502328]And this time, I'm not aware of any leaks from the server. But then again, I haven't gone looking. :smile:[/QUOTE]Because you know the exponent. Did your machine beep loudly?

 Mark Rose 2018-12-11 10:43

[QUOTE=Uncwilly;502344]Because you know the exponent. Did your machine beep loudly?[/QUOTE]

I do not know the exponent for this newly discovered prime.

At the time of the 77232917 discovery there was a leak in the xml report for the exponent. MadPoo quickly fixed that though. :smile:

 Dr Sardonicus 2018-12-11 14:45

[QUOTE=chalsall;502335]Yeah. Be very careful what you do (and take) after an injury.

Sometimes your Doctors will be financially motivated to prescribe to you some seriously addictive stuff.

Cocaine nor Nicotine nor Caffeine have anything like the effects some doctors are regularly prescribing now-a-days.[/QUOTE]My parents' doctor said that morphine was good for relieving post-surgery pain -- [i]for about three days![/i] He didn't want [i]his[/i] patients getting hooked on the stuff.

I distinguish between analgesics (aspirin, ibuprofen, naproxen, acetaminophen) and anesthetics ("painkillers") -- central nervous system depressants like opioids or alcohol.

If ice, rest, and ibuprofen give satisfactory results, stick to it. You probably don't want to overdo it on the ibuprofen, though. Not good for your stomach.

In my experience, there are some kinds of acute joint pains that ice and analgesics don't seem to touch, but on which alcohol has an almost miraculous relieving effect. A stiff drink after supper -- no more than two nights in a row -- might be helpful if ice and analgesics are giving unsatisfactory results. My theory is, it dulls the pain and unclenches your muscles, so you sleep more soundly, allowing your own body to do its healing work. Beyond a couple of days, though, the deleterious effects of alcohol will probably outweigh the surcease of pain from its anesthetic embrace. You also don't want to wash down a lot of ibuprofen with alcohol. [i]Very[/i] bad for your stomach.

 Prime95 2018-12-11 18:03

And now the part y'all are going to hate...

I plan on announcing after I return from the cruise. I get home the night of the 19th. I can't see getting everything ready until the 21st. Yet, Fridays are supposed to be bad days for announcing news, but the next week everyone takes off. What to do??? Take chances with a Friday announce, wait to the Christmas week, wait for the new year???

The downside of "lucky 13" is coming up with a new "spin" for the main page blurb.

 chalsall 2018-12-11 18:35

[QUOTE=Prime95;502414]The downside of "lucky 13" is coming up with a new "spin" for the main page blurb.[/QUOTE]

What about something along the lines of "Yes, Virginia, There is a Santa Claus

Try working in the (somewhat esoteric) angle that this discovery was quite unexpected (statistically)?

Just throwing out ideas...

For those who haven't been involved with the timing of the announcements, it's a bit of an art (and a whole lot of guessing (and luck)). Announce on a "big news day" and no one notices; announce during a "slow news day" and things go wild!!!

 Mark Rose 2018-12-11 19:07

Friday the 21st would be particularly bad, since many people will also have Monday the 24th off and will be distracted. I'd wait until the 27th.

 Uncwilly 2018-12-11 19:40

[QUOTE=Prime95;502414]What to do??? Take chances with a Friday announce, wait to the Christmas week, wait for the new year???

The downside of "lucky 13" is coming up with a new "spin" for the main page blurb.[/QUOTE][QUOTE=chalsall;502417]What about something along the lines of "Yes, Virginia, There is a Santa Claus

Try working in the (somewhat esoteric) angle that this discovery was quite unexpected (statistically)?[/QUOTE][QUOTE=Mark Rose;502421]Friday the 21st would be particularly bad, since many people will also have Monday the 24th off and will be distracted. I'd wait until the 27th.[/QUOTE]
Since Boxing day is the 1 year anniversary of the previous discovery it might be better to drop the announcement then and play up the anniversary angle in addition to the good ideas that Chris had posted. Not waiting until Jan is hopefully advisable. End of December is a time for feel good stories. Play up that aspect.

Also, this maybe be enough evidence to suggest that our current understanding is incorrect. Is the current trend indicative of something different or is this randomness being clumpy? Help GIMPS find out.

 JeppeSN 2018-12-11 22:05

[QUOTE=ewmayer;502428]I don't think "clustered" is the right word ... the last 12 simply appear to lie on a different-sloped regression line than the previous ones, which more or less followed the Wagstaff-heuristic prediction. (I believe I can post this log-plot w/o giving away too much re. the latest find ... but in any event we already have 1 independent-hw/sw confirmation, so it's not a false positive).[/QUOTE]

I think I can measure the position, with two decimals, on the ordinate axis of that graph, its last dot. I get 18.xy where x and y are two digits I know. Then the prime exponent p we seek, is approximated by p = exp(18.xy) where exp is the natural exponential function. /JeppeSN

 ewmayer 2018-12-11 22:38

[QUOTE=JeppeSN;502439]I think I can measure the position, with two decimals, on the ordinate axis of that graph, its last dot. I get 18.xy where x and y are two digits I know. Then the prime exponent p we seek, is approximated by p = exp(18.xy) where exp is the natural exponential function. /JeppeSN[/QUOTE]

Note that +-0.01 in terms of ln(x) for ln around that of the previous find would narrow the exponent to within a million or so, a bit narrower than ATH's previous reveal. It's nice to give the data-miners hereabouts a few clues on the way to the official announcement. :)

 Mark Rose 2018-12-11 23:59

[QUOTE=Madpoo;502236]The guessing game has a long history. In a way, it's half the fun. I think we've done a fair job of keeping details hidden on the server itself. In years past it was pretty obvious that a prime was found because the report page would show a success in a particular 1M range. LOL[/QUOTE]

Though the numbers on that page don't add up. If you take the Range Count column, subtract the two Composite columns, the Assigned, and the Available, the 86M range has 4 unaccounted for exponents, and the 89M range has 2.

 GP2 2018-12-12 00:35

[QUOTE=Prime95;502414]And now the part y'all are going to hate...

I plan on announcing after I return from the cruise. I get home the night of the 19th. I can't see getting everything ready until the 21st. Yet, Fridays are supposed to be bad days for announcing news, but the next week everyone takes off. What to do??? Take chances with a Friday announce, wait to the Christmas week, wait for the new year???

The downside of "lucky 13" is coming up with a new "spin" for the main page blurb.[/QUOTE]

One PR idea: ATH used a Tesla V100 to do the verification in record time. I don't think we've ever had an Mersenne prime verified so quickly. From [URL="https://mersenneforum.org/showthread.php?p=502199&postcount=16"]2018-12-09 20:09[/URL] to [URL="https://mersenneforum.org/showthread.php?p=502288&postcount=41"]2018-12-10 15:50[/URL], or less than 20 hours at most.

I presume he used a p3 instance in the AWS cloud? Maybe [URL="https://aws.amazon.com/blogs/aws/author/jbarr/"]Jeff Barr[/URL] at the AWS News Blog could spin a blog entry out of it, who knows.

Maybe Amazon or nVidia might even want to play up the "when you need to get results fast!" angle.

 xx005fs 2018-12-12 00:57

[QUOTE=Prime95;502167]On Dec 7th, a new Mersenne prime was reported. I emailed the discoverer, obtained the last save file, and reran the last 100,000 iterations. Sure enough, we have a new prime!

Aaron has started a full test using prime95, but that doesn't count as an official double-check. We need a volunteer to run CudaLucas and/or mlucas.

As usual, the automatic email notification to myself and past Mersenne prime discoverers failed. Yes, the automatic email was debugged and tested after its last failure. Apparently bit rot set in. Fortunately the backup notification plan worked.[/QUOTE]

Congrats!!

 xx005fs 2018-12-12 01:00

[QUOTE=GP2;502455]One PR idea: ATH used a Tesla V100 to do the verification in record time. I don't think we've ever had an Mersenne prime verified so quickly. From [URL="https://mersenneforum.org/showthread.php?p=502199&postcount=16"]2018-12-09 20:09[/URL] to [URL="https://mersenneforum.org/showthread.php?p=502288&postcount=41"]2018-12-10 15:50[/URL], or less than 20 hours at most.

I presume he used a p3 instance in the AWS cloud? Maybe [URL="https://aws.amazon.com/blogs/aws/author/jbarr/"]Jeff Barr[/URL] at the AWS News Blog could spin a blog entry out of it, who knows.

Maybe Amazon or nVidia might even want to play up the "when you need to get results fast!" angle.[/QUOTE]

The Volta series card (Titan v or v100) are definitely the fastest GPU for LL testing, also the most expensive of the bunch (i guess titan v is cheaper than p100?). 20 hours even seems a bit slow for a v100 at the expected number range imo. If the electricity price is very high and you have deep pocket, Titan V is the best value LL cruncher for sure while using very little amount of power compared to lots of CPU, also much more expensive.

[QUOTE=ewmayer;502428]I don't think "clustered" is the right word ... the last 12 simply appear to lie on a different-sloped regression line than the previous ones, which more or less followed the Wagstaff-heuristic prediction. (I believe I can post this log-plot w/o giving away too much re. the latest find ... but in any event we already have 1 independent-hw/sw confirmation, so it's not a false positive).[/QUOTE]

I see what's happening in that plot... they're approaching a limit and we'll probably see a bunch more in the current range, and then that's it, no more primes, ever. LOL (yes, I'm kidding, but the curve did remind me of something akin to an asymptote).

 Batalov 2018-12-12 03:09

[QUOTE=ATH;475537]Btw the corresponding Perfect number +1 has small factors:
2[sup]p-1[/sup]*(2[sup]p[/sup]-1)+1

and the corresponding Wagstaff number has a factor (NMC):
2[sup]p+1[/sup] / 3

Still no factor of 2[sup]p[/sup] - 3 (twin prime)[/QUOTE]

[QUOTE=chalsall;502443]Antialiasing...

Just to put on the table, I wish that people "in the know" would refrain from dropping hints. Many times in the past errors have been made (sometimes in combination with other hints) which narrowed down the possibilities to be far too fine for comfort....[/QUOTE]

Agreed... there's "dropping hints" and there's "dropping hints".

I was thinking about other hints that could be dropped, like it's congruent to X mod 18 or whatever, and then I took the time to crunch the numbers to see if doing so, in confluence with the other hints, would narrow it down *too* much.

I realized that if I take the hints out there and then make some reasonable assumptions, like being between 82M - 87M since that's the approx range being handed out, and then taking into account that a lot of exponents in there are unassigned still, I realized that dropping just one more clue like a MOD would narrow it down to about a few dozen. LOL

I was looking at the decimal digits of the prime and looking for fun patterns, like runs of the same digit in a row, or how many significant digits of pi I could find, etc. Even sharing that would lead some enterprising folks to the exponent because you could do the calculation with yafu or something and search it 'til you find one that matched those things, and it wouldn't take too much time per exponent. :smile:

 chalsall 2018-12-12 04:03

[QUOTE=Madpoo;502469]Even sharing that would lead some enterprising folks to the exponent because you could do the calculation with yafu or something and search it 'til you find one that matched those things, and it wouldn't take too much time per exponent. :smile:[/QUOTE]

Yeah. It's a bit like a difficult Sudoku, for extreme geeks.

While the language is a bit weak, may I please remind everyone of the [URL="https://www.mersenne.org/legal/"]GIMPS Legal[/URL] document. Specifically:

[QUOTE][U]Discovery Non-Disclosure Period.[/U] Participants and Awardees agree to a reasonable immediate period of non-disclosure communications silence, not to exceed 90 days, about any discovery, alleged or verified, until GIMPS announces the discovery to the academia and press. Non-compliance may result in award disqualification.[/QUOTE]

At the end of the day it's not really enforceable (or, at least, there's little downside for non-compliance).

But the intention is that it is George's right to make the announcement when and how he chooses (within a reasonable amount of time).

 paulunderwood 2018-12-12 07:42

Are we there yet? :xmastree:

 JeppeSN 2018-12-12 18:17

Careful what bases we uses with the logarithms. We have:
[$$]\log_b x = \frac{1}{\log_a b}\log_a x[/$$]
which means the different logs are proportional (the fraction is a constant).

Wagstaff says the number of prime [$]M_p[/$] below [$]x[/$] is about [$]\frac{e^\gamma}{\log_e 2}\log_e \log_e x[/$].

We have [$]p = \log_2 M_p[/$] (we just disregard the [$]-1[/$] term here). If you take "common" log of that, you look at [$]\log_{10} \log_2 x[/$] rather than [$]\log_e \log_e x[/$], so you need to adjust the slope with a constant. See?

/JeppeSN

 ATH 2018-12-12 20:06

MLucas run just finished:

M8xxxxxxx is a new MERSENNE PRIME!!! :beer2: :beer2: :beer2:

So we have triple hardware tests: Core ix, Tesla V100, Xeon

and triple software tests: Prime95, CUDALucas, Mlucas.

 chalsall 2018-12-12 20:09

[QUOTE=ATH;502536]So we have triple hardware tests: Core ix, Tesla V100, Xeon and triple software tests: Prime95, CUDALucas, Mlucas.[/QUOTE]

Sweet! :tu:

 lycorn 2018-12-12 20:50

We've done it again!
Congrats everybody.

 Gordon 2018-12-12 22:53

[QUOTE=chalsall;502443]Antialiasing...

Just to put on the table, I wish that people "in the know" would refrain from dropping hints. Many times in the past errors have been made (sometimes in combination with other hints) which narrowed down the possibilities to be far too fine for comfort....[/QUOTE]

I'm saying nothing :smile:

[QUOTE=chalsall;502516]Why not simply wait for the official announcement?

IMO, this leak (the original PDF) was ill-advised....[/QUOTE]

Well, one more hint. The prime is odd, not even. :smile:

That's especially humorous to me because when I used yafu to generate the file for the decimal digits of the new prime, I forgot to subtract one. Yes, I did 2^p and not 2^p-1

So there's my huge file of 24-25 some odd megabytes, and I'm looking through it to see various quirks in the data, and I notice the last digit is even, not odd. Major whoopsie. And I'd already formatted it to 100-character width, zipped it up... man that's embarrassing.

I'm pretty sure I did it right the next time but I even stopped to compare and made sure the only diff was that last digit or two (see... I'm not even giving away if the last digit is a 9 by saying maybe the last 2 changed).

Which reminds me, I think last time, someone mentioned the last digit, which really helped narrow it down. I don't think it was long before people just trivially looked at the exponents they were already looking at, go the last digit, and I do remember seeing someone guess the correct one in that thread. Well, maybe they would have got lucky anyway.

 JeppeSN 2018-12-13 12:40

[QUOTE=Madpoo;502574]and made sure the only diff was that last digit or two (see... I'm not even giving away if the last digit is a 9 by saying maybe the last 2 changed).[/QUOTE]

I think it safe to say that a power [$]2^p[/$] of two, cannot end in a digit zero, because the prime 5 cannot divide a power of the prime 2. Therefore, when you subtract one from a power of two, only the last decimal digit changes.

Besides, they already gave us the exponent modulo 4, so we can infer that your correction consisted in replacing a final digit 2 with a 1.

/JeppeSN

 JeppeSN 2018-12-13 15:33

[QUOTE=Dr Sardonicus;502605]I think you mean, 85000000 for the [i]exponent[/i], which is (to a fare-thee-well) for a reasonably large Mersenne prime x, p = log(x)/log(2), so that log(x) = p*log(2). Hmm. I'd better check. I have an unfortunate tendency to put terms on the wrong side of the fraction bar.

? exp(Euler)*log(85000000*log(2))/log(2)

%1 = 45.973385236161997342391226401795683851

OK, looks good.

I vote for "haven't reached asymptotically' yet." The exponents haven't even reached 10[sup]8[/sup]. My analytic number theory prof once said, "Analytic number theory begins at 10[sup]40[/sup]."

What's [i]my[/i] guess for the new exponent? I don't have one. I reckon, I'll know soon enough...[/QUOTE]

Yeah, if you try with 10M, that is:

(exp(Euler)/log(2))*log(log(2.0^10e6))

you get 40.47, which is too much (there are only 38 Mersennes there in reality). The, as you say, with 85M:

(exp(Euler)/log(2))*log(log(2.0^85e6))

it says 45.97 which is too little, since we know there are 51 or more.

/JeppeSN

 mvusse 2018-12-13 16:01

[QUOTE=JeppeSN;502598]
Besides, they already gave us the exponent modulo 4, so we can infer that your correction consisted in replacing a final digit 2 with a 1.
/JeppeSN[/QUOTE]

Umm aren't 9, 13 and 17 also 1 modulo 4, making the last digit either 1, 3, 7 or 9?

 GP2 2018-12-13 16:36

[QUOTE=R. Gerbicz;502593]So we have roughly 24 percentage to see at least one crowd, nothing that very especial.[/QUOTE]

You are probably right.

But every time we convince ourselves of that, someone goes and discovers yet another Mersenne prime, and then the speculation starts all over again.

Maybe if we look hard enough for coincidences, we find them everywhere, like "[URL="https://en.wikipedia.org/wiki/Six_nines_in_pi"]six nines in pi[/URL]".

 chalsall 2018-12-13 16:50

[QUOTE=Madpoo;502574]I don't think it was long before people just trivially looked at the exponents they were already looking at, go the last digit, and I do remember seeing someone guess the correct one in that thread. Well, maybe they would have got lucky anyway.[/QUOTE]

Yeah... And further, someone edited the entry at Wikipedia about MPs listing the 50th MP, but they had guessed incorrectly! This was fortunately noticed quickly and reverted, but it could have diluted the authority of the official announcement.

Again, let me [B]please[/B] implore people to let George et al announce officially. If done correctly it can result in a huge increase in throughput for GIMPS.

Admittedly the increase is a surge which then tapers off as the news reports do, but some newly acquired participants do stick around. Further, because of the new assignment rules, this should help decrease the gap between the DC and LL wave-fronts.

[QUOTE=chalsall;502632]Yeah... And further, someone edited the entry at Wikipedia about MPs listing the 50th MP, but they had guessed incorrectly! This was fortunately noticed quickly and reverted, but it could have diluted the authority of the official announcement.

Again, let me [B]please[/B] implore people to let George et al announce officially. If done correctly it can result in a huge increase in throughput for GIMPS.

Admittedly the increase is a surge which then tapers off as the news reports do, but some newly acquired participants do stick around. Further, because of the new assignment rules, this should help decrease the gap between the DC and LL wave-fronts.[/QUOTE]

Plus, the new version of Prime95 stores interim partial residues along the way. Although we're not currently doing anything with them, if someone starts but doesn't finish a test, we may still be able to use those interim residues to help gauge whether a completed test seems accurate or not. If they diverge at any point, I'd say those were candidates for the early double-checking project.

 Prime95 2018-12-13 18:17

[QUOTE=Dr Sardonicus;502605]
I vote for "haven't reached asymptotically' yet." .[/QUOTE]

Me too. Hard to believe that dealing with numbers of 20+ million digits we haven't reached asymptotic yet. But, "log log" is a powerful "reducer".

The difference between asymptotic behavior and current behavior is the "a" in the "log ap" calculation (where a is 2 or 6 and p is the exponent). If p is large, a can be ignored. Right now, log ap = log a + log p, and log p is about 18 and log 2 is 0.7 and log 6 is 1.8. Thus we should be finding new Mersenne primes about 4% or 10% more often than the asymptotic behavior. This well explains my simulation getting 50.5 hits vs. the asymptotic expected 46.

The section of Wagstaff's paper where he compares the slope of known Mersenne primes in 1983 to his asymptotic formula for expected number of Mersenne primes is pretty much rubbish as back then we were far far away from asymptotic behavior. I think Chris's web page could also use some modifications.

My conclusion (and I am NOT a mathematician): Wagstaff-Lenstra-Pomerance formula is still looking good (from the perspective of number of Mps found). We've been on a lucky streak that has reverted us to the expected norm.

Now, back to drinks by the poolside.

 Dr Sardonicus 2018-12-13 18:55

[QUOTE=GP2;502629]You are probably right.

But every time we convince ourselves of that, someone goes and discovers yet another Mersenne prime, and then the speculation starts all over again.

Maybe if we look hard enough for coincidences, we find them everywhere, like "[URL="https://en.wikipedia.org/wiki/Six_nines_in_pi"]six nines in pi[/URL]".[/QUOTE]
A couple of fun things about the number pi; one related to Feynman, the other to whether pi is a "normal number" (both of these being mentioned in the above link):

From [u]Surely You're Joking, Mr. Feynman![/u] ("Lucky numbers"):[quote]One day I was feeling my oats. It was lunch time in the technical area, and I don't know how I got the idea, but I announced, "I can work out in sixty seconds the answer to any problem that anybody can state in ten seconds, to 10 percent!" People started giving me problems they thought were difficult, such as integrating a function like 1/(1 + x[sup]4[/sup] ), which hardly changed over the range they gave me. The hardest one somebody gave me was

the binomial coefficient of x[sup]10[/sup] in (1 + x)[sup]20[/sup]; I got that just in time.

They were all giving me problems and I was feeling great, when Paul Olum walked by in the hall. Paul had worked with me for a while at Princeton before coming out to Los Alamos, and he was always cleverer than I was.

...

So Paul is walking past the lunch place and these guys are all excited. "Hey, Paul!" they call out. "Feynman's terrific! We give him a problem that can be stated in ten seconds, and in a minute he gets the answer to 10 percent. Why don't you give him one?" Without hardly stopping, he says, "The tangent of 10 to the 100th." I was sunk: you have to divide by pi to 100 decimal places! It was hopeless.[/quote]

There's a passage in a book called [u]The Lore of Large Numbers[/u] by Philip J. Davis, in a chapter about the number pi, which mentions a question dreamed up by the great mathematician L. E. J. Brouwer. His question was, whether in the decimal expansion of pi there were ever a thousand consecutive zeros. AFAIK the answer is not known. But if pi is a "normal number" to the base ten, then such a block of digits would occur with frequency abouyt 1 in 10[sup]1000[/sup].

 JeppeSN 2018-12-13 23:33

[QUOTE=mvusse;502619]Umm aren't 9, 13 and 17 also 1 modulo 4, making the last digit either 1, 3, 7 or 9?[/QUOTE]

If you look at this table:
[$$]\begin{matrix} p: & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \dots \\ 2^p: & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512 & 1024 & \dots \end{matrix}[/$$]
you will notice that the last digit goes like [$]2,4,8,6,2,4,8,6,2,4,\ldots[/$]. Every fourth final digit is a [$]2[/$]. In fact, if you disregard [$]M_2=2^2-1=4-1=3[/$], every Mersenne prime will be one less than a number ending in either [$]2[/$] or [$]8[/$], because only the odd exponents [$]p[/$] matter. /JeppeSN

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