Doubts, scientific breakthroughs and conspiracy theories about Wagstaff Conjecture (from Lucky13)

[Batalov]

It is interesting that Pomerance, Lenstra and Wagstaff looked for heuristics when MP set was only known up to p<100,000. (ca. 1983)


Might there be some better summation techniques for the excluded factor probability corrections now? Those look pretty rough in the 1983 heuristic. E.g. the summation stops at 2k or 6k, but there are many more excluded divisors whose contributions to the correction alone are increasingly small but as a mass they can contribute significantly.

[kriesel]

Originally Posted by Prime95

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.

What's 13 got to do with it?

[chalsall]

Originally Posted by JeppeSN

I still cannot believe these numbers occur more frequently than conjectured.

Why? The empirical feeds into the theory; and tends to be taken more seriously in the analysis.

Belief is an artefact of presuppositions.

[The Carnivore]

Originally Posted by kriesel

What's 13 got to do with it?

It may have been the user's 13th LL test.

[Prime95]

Originally Posted by kriesel

What's 13 got to do with it?

The 13th Mersenne prime between 10,000,000 and 100,000,000 -- expected 6.

For comparison:
1 to 10: 4
10 to 100: 6
100 to 1000: 4
1000 to 10000: 8
10000 to 100000: 6
100000 to 1000000: 5
1000000 to 10000000: 5

[ET_]

How (im)probable is to find such a cluster of Mersenne primes grouped together instead of laying on the linear regression line?

[GP2]

Originally Posted by Prime95

1 to 10: 4
10 to 100: 6
100 to 1000: 4
1000 to 10000: 8
10000 to 100000: 6
100000 to 1000000: 5
1000000 to 10000000: 5

And for an example of what a real drought looks like, consider the Wagstaff primes.

There is most likely only one prime in the 1 million to 10 million range, pending a verification run which has reached 6 million so far.

[ewmayer]

Originally Posted by ET_

How (im)probable is to find such a cluster of Mersenne primes grouped together instead of laying on the linear regression line?

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).

[GP2]

Originally Posted by JeppeSN

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

Assuming the plot is accurate, analysis of the pixels places the new exponent between 82.60M and 82.75M 82.65M.

We could do better than that of course. Simply identify the software that created the plot and start plotting points, varying the exponent value until you come up with a pixel-by-pixel reproduction of the plotted box pattern, including the pale shading and blur (sorry, I don't know the correct graphics terminology).

[chalsall]

Originally Posted by GP2

...shading and blur (sorry, I don't know the correct graphics terminology).

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....

[ewmayer]

I would be interested in some discussion of the statistical significance ascribable to the apparent trendline break of the last dozen M-prime exponents. Chris Caldwell's "this graph is amazingly linear" is getting more untenable with each new sooner-than-expected find.