Predict M52

[Spherical Cow]

Originally Posted by Uncwilly

Other than those, the following have had their prediction date passed (and are eligible for re-guessing per rule 6):

Who		When guessed	Exponent	When found
dcheuk  	3/11/2019	87668491	3/14/2019
kriesel  	12/21/2018	88318673	6/14/2019
christian_	12/27/2018	93456799	4/7/2019
Spherical Cow	12/27/2018	97211981	10/30/2019
dcheuk  	9/1/2019	333898333	11/1/2019

As a member of the above elite group, my revised prediction is 107448791, which will be discovered on February 9, 2021.

Norm

[Uncwilly]

Originally Posted by Uncwilly

84000017 on Feb 28, 2019

This one is well passed, so I will replace it with:

99998441 on July 31, 2020

[dcheuk]

Originally Posted by Uncwilly

...
Other than those, the following have had their prediction date passed (and are eligible for re-guessing per rule 6):

Who		When guessed	Exponent	When found
dcheuk  	3/11/2019	87668491	3/14/2019
kriesel  	12/21/2018	88318673	6/14/2019
christian_	12/27/2018	93456799	4/7/2019
Spherical Cow	12/27/2018	97211981	10/30/2019
dcheuk  	9/1/2019	333898333	11/1/2019

...

Reguess:
98350193
June 1st, 2020

Kind of surprised still no prime found yet.

[kriesel]

Originally Posted by dcheuk

Kind of surprised still no prime found yet.

Why? Best heuristics say there are only about 6 left to find in p<10⁹, which will take many decades to search completely. https://www.mersenneforum.org/showpo...5&postcount=11

https://www.mersenne.org/primes/ contains many waits of 5+ years, and some longer than a century.

[Uncwilly]

Originally Posted by kriesel

https://www.mersenne.org/primes/ contains many waits of 5+ years, and some longer than a century.

Anything before Robinson is completely irrelevant to our current search. Some would argue that anything before GIMPS started is also not relevant to the length of time between discoveries.

Based upon the gaps between primes (relative to the size the primes), masser's guesses are close to the "expected" value.

[kriesel]

Originally Posted by Uncwilly

Anything before Robinson is completely irrelevant to our current search. Some would argue that anything before GIMPS started is also not relevant to the length of time between discoveries.

Based upon the gaps between primes (relative to the size the primes), masser's guesses are close to the "expected" value.

The GIMPS era includes a 3.7 year wait. And it is likely to include much longer waits in the future. The supercomputer era includes a 7 year wait. The search always becomes rate limited by the available computing power, whatever that technology is. Transitions in what technology or quantity is available sort themselves out, relatively quickly. The actual spacing on the number line of Mersenne primes is individually unpredictable but overall predictable. See https://primes.utm.edu/notes/faq/NextMersenne.html. The location of Mersenne primes on the number line is innate, it is not affected by the technology or quantity of computing resources. Given that the computational effort of one primality test scales as p^2.1, using the best available algorithms, and on the average p_n/p_n-1 ~1.476, effort ramps up quickly per primality test, and more primality tests are needed also. (Probable interval grows proportional to p, while density of prime exponents in the interval drops more slowly, by log(p).) I recall seeing the combined effect of those considerations expressed on this forum before as on the average, finding the next Mersenne prime is EIGHT times the effort of finding the previous. The increased effort per higher exponent is why maintaining even a constant rate of minor milestones per year is increasingly more challenging. See https://www.mersenneforum.org/showth...995#post502995

We have had an unusually lucky run lately in the 2016 through 2018 period. The ratio of exponents of apparently successive Mersenne primes were improbably close together recently. To expect that to continue is not justified. A continuation of such close spacing of primes is counter to number theory, probability, and historical experience. See the examples of 100-1000 (>4:1), and 100K-1000K (>3:1). The 1M-20M stretch contains 3 cases where p_n/p_n-1 > 2. https://www.mersenne.org/primes/

If we happen to now be in a ~2:1 step, at our recent rate of advancing primality first-test milestones, 6M/year, we would have about a 10-13 year remaining wait depending on assignment and completion timing luck at the wavefront. (166-103)/6 ~ 10.5 vs (166-89)/6 ~ 12.8
I hope that is not the case. But I know of no good theoretical or empirical reason to expect not to encounter such a case or larger ratio at some point. There are good reasons to expect droughts to happen again.

[chalsall]

Originally Posted by kriesel

We have had an unusually lucky run lately in the 2016 through 2018 period. The ratio of exponents of apparently successive Mersenne primes were improbably close together recently. To expect that to continue is not justified.

But... Iff we do find another sooner rather than later, it will be a /really/ interesting datapoint for the theorists.

The maths for why is /way/ beyond my abilities. But it's important enough to have been mentioned in the last MP announcement.

[R.D. Silverman]

Originally Posted by chalsall

But... Iff we do find another sooner rather than later, it will be a /really/ interesting datapoint for the theorists.

Negative.

The maths for why is /way/ beyond my abilities. .

Yet you presume to make unjustified pronouncements. What compels you
to do so?

[Uncwilly]

Originally Posted by R.D. Silverman

Negative.

How many data points coming sooner than currently expected would it take for you to think that it is significant? 10, 20, 50, 100?

Yet you presume to make unjustified pronouncements. What compels you to do so?

Because it is fun and this thread is about fun and not hardcore math. Some people like to have fun like this.

[kriesel]

None of the following should be taken as disparaging to anyone's posts or enthusiasm. Only as encouragement to be accurate, and realistic.
And the ratio 1.47576 is only the estimated mean, about which large variations are expected to occur very frequently.

Uncwilly summarized pending guesses at https://mersenneforum.org/showpost.p...4&postcount=87 on Dec 19 2019, including
masser 8/3/2019 115856161 7/4/2023
masser 12/12/2018 117003179 11/24/2022
masser 12/12/2018 118788157 8/8/2020

In https://mersenneforum.org/showpost.p...&postcount=104 Uncwilly stated

Based upon the gaps between primes (relative to the size the primes), masser's guesses are close to the "expected" value.

It's not clear to me exactly what that means, but suppose it means relative to a recent prime discovery such as 77232917. 1.47576 times larger would be ~113977249, which is prime. From where we're at now, the first-primality-testing wavefront is years from reaching that. The middle of that wavefront is now at ~99M, and last year the first-test milestone increased by 5M, so we are about 3 years away from it, and about 4 years away from masser's highest guess. (From 82589933, it would be ~121882919; 121882903 is the nearest smaller prime. That's more than a year further off at our recent aggregate rate of progress, or 5+ years away.)

Some of us will find any additonal prime discoveries interesting, regardless of their positions.
We are happy to provide such additional data to the number theorists, who will make of them whatever they will.
It would take a lot of such new finds, or a very wide span of their absence, to raise serious doubt about the current conjectures regarding their distribution. Which will probably take lifetimes to achieve, given the rate of computing throughput increase in GIMPS and the size of the task.

There are numerous threads which have been placed into misc math which make various dubious claims about particular exponents being Mersenne primes. Most of those have already been refuted by the facts developed by targeted factoring and primality testing. Those remaining unrefuted so far, that are amenable to primality testing with current software and hardware, are expected to be resolved in 2020. Not all of us find leaving unrefuted unjustified claims laying about unchallenged to be fun.

There is a balance to be struck, in the less formal threads, between accuracy and rigor on the one hand, and fun on the other. And people will differ on what the right balance is.

[Uncwilly]

Originally Posted by kriesel

It's not clear to me exactly what that means, but suppose it means relative to a recent prime discovery such as 77232917. 1.47576 times larger would be ~113977249, which is prime.

I did my own calc. I ignored primes below 127 as the number of digits is too small to be useful (too noisy).
I did the following:
calculated log₁₀(digits in MP_x) for each MP
took difference between successive values
took the mean
added the mean to the value calculated for 82589933 (M51)
converted the resultant value back into the value for p

I got ~117,xxx,xxx Very close to the middle of masser's guesses.

It is all fun and games until someone gets serious.