was M40 predicted?

[tha]

L.S.,

In the welcome thread of the Lone Mersenne Hunters Forum a reference is made to the prediction of narrow margins in which Mersenne Primes are expected on statistical grounds.
One prediction has proven to be false, but others were frightning precise. There is no prediciton of a Mersenne Prime in the 20M range mentioned, since a the posting was made long time ago. But from the numbering it can be concluded that there may very well have been a prediction. It would be very interesting to know which region was predicted. Eric?

[GP2]

Originally posted by tha
One prediction has proven to be false, but others were frightning precise.

Not exactly "proven" false:

M#39 - 53.7390% probability - range=10987349-11013853
M#39 - 64.0127% probability - range=10914203-11092621
M#39 - 81.6073% probability - range=10793527-11204183
M#39 - 97.3391% probability - range=10526447-11390453

We're not finished double-checking this range.
Who knows, maybe one of the exponents in the recently re-released batches will turn up something yet.

[pakaran]

Originally posted by GP2
Not exactly "proven" false:

M#39 - 53.7390% probability - range=10987349-11013853
M#39 - 64.0127% probability - range=10914203-11092621
M#39 - 81.6073% probability - range=10793527-11204183
M#39 - 97.3391% probability - range=10526447-11390453

We're not finished double-checking this range.
Who knows, maybe one of the exponents in the recently re-released batches will turn up something yet.

How were these numbers reached? I can't imagine predicting a Poisson distribution that precisely. Some real world distributions you CAN narrow down (most likely an adult man will be between 5'4 and 6'2), but not Mersenne primes, which have a habit of being anywhere from the fourth power of their predecessor (521) down to a roughly 2% margin in exponent size (the 3M twins).

I will use the classical CS abbreviation and make lg the base 2 log.

lg lg M39 is slightly greater than 22.7332. This value divided by 39 (to split it evenly between 1 and the first 39 primes) is .582904. If we add this to lg lg(M13466917) = 23.6829, we would expect the next Mersenne to be 2^2^24.266 (I drop a significant digit to be paranoid), or 20.172 million.

This is about what we would estimate from previous discussion on this forum. Note though that estimating up from previous Mersennes gives estimates that are significantly off from this answer.

So the bottom line is I'm not sure whether statistical arguments can locate Mersenne primes to within tolerances of 5% in exponent (a lg lg difference of 0.070389).

Using lg lg PRIME is useful for these kinds of discussions because it lets the expected locations be estimated by mental arithmetic, and found quite precisely with a scientific calculator.

[GP2]

Originally posted by pakaran
How were these numbers reached? I can't imagine predicting a Poisson distribution that precisely.

The original mailing list message was here:
http://www.mail-archive.com/mersenne.../msg05046.html

As you can see, he refused to reveal anything about his methods.

[GP2]

See
http://www.utm.edu/research/primes/n...tMersenne.html
for a graph and discussion of what pakaran was talking about.

See also
http://opteron.mersenneforum.org/png/log2_P_vs_N.png
for the same graph.

[GP2]

Originally posted by pakaran
lg lg M39 is slightly greater than 22.7332. This value divided by 39 (to split it evenly between 1 and the first 39 primes) is .582904.

Wagstaff's conjecture is that this slope is
1/e^gamma = 0.56145948...
where gamma is Euler's constant.

See the link in the previous post.

PS,
actually it's lg lg M38 that is = 22.7332...

lg lg M39 = 23.6829...