Predict M45...

[Fusion_power]

distribute it among fewer numbers (the candidates that survived the TF)

In the case of numbers currently being tested, trial factoring does not reduce the total number of potential factors by any significant amount. As numbers grow larger, the efficiency of trial factoring decreases though not at a linear rate because larger numbers also have larger 2kp+1 values of potential factors.

One of the reasons why searching for Mersenne primes is interesting is because there is no way currently known to predict which number will be prime. Statistics can give a hint of how many should be in a given region but that is all they do.

Fusion

[cheesehead]

Originally Posted by davieddy

But if a trial factor isn't found, the exponent is more likely to yield a prime.

... "more likely" in the sense of our a priori expectation probability for that exponent, not the actual (a posteriori) likelihood, which never changes and is either 0 or 1. (Just to be clear^H^H^H^H^H ... picky. )

What I am suggesting is that trial factoring a range of exponents has a negligible effect on the expected number of Mersenne primes in the range.

... negligible for the reason that the feasible amount of trial factoring is a negligible fraction of the theoretical range (all the way to the square root).

As we TF (and P-1 and ECM and ..., but not L-L) more and more, a more complete (i.e., accounting for the extent of factoring effort) calculation of a priori expectation of primes in a range will converge toward the actual number of Mersenne primes! Slowly and not monotonically, but surely.

After all, if we were able to actually TF all the way to the square roots, then as the TF progressed higher our expectation of each exponent would rise toward 1 until either a factor were found, at which time the expectation would suddenly drop to zero, or until we reached the square root, at which time the expectation would reach 1.

Originally Posted by ewmayer

TF depth, since the only effect of the latter is to take the summed probability of of the interval in question yielding a prime and distribute it among fewer numbers (the candidates that survived the TF).

I'd prefer to say that TF depth has no effect at all on George's current calculation except the indirect one of reducing, over time, the number of unfactored candidates over which George's method calculates the summed probability.

[petrw1]

Well historically there has been the following distribution of Mersenne Primes:
10^0 digits: 7
10^1 digits: 5
10^2 digits: 6
10^3 digits: 8
10^4 digits: 5
10^5 digits: 6
10^6 digits: 7
I would say the distribution is fairly normal (dare I say predictable).
Based on this I would suggest the next 5-8 will have 10^7 digits: between M33219279 and about M332192800.

The smallest MP in the last few ranges has started with a 2. Continuing that trend I will suggest more accurately that M45 will be around M61000000....discovered on my birthday: April 3, 2011.

[Damian]

Never. All mersenne primes have already been found.
Any proof I'm wrong?

[Mini-Geek]

Any proof you're right?

[jasong]

Has anyone predicted that M44 is actually M45? If not, I'll claim that. (Of course it'll be somewhere around 2010 before we know I'm wrong)

[petrw1]

The trend for the last 10 prime findings has been very near liinear ... an average of 13 months between findings ... so I predict the next MP will be found Oct 3, 2007 (my Math geek son's birthday)

[davieddy]

Originally Posted by petrw1

The trend for the last 10 prime findings has been very near liinear ... an average of 13 months between findings ... so I predict the next MP will be found Oct 3, 2007 (my Math geek son's birthday)

In that case we could narrow down the likely value of the exponent
quite a bit. Let's say 37 or 38 million or so. That's where most of the
completed tests will be coming from.

[masser]

M51360853 mid 2009

[brunoparga]

Originally Posted by jasong

Has anyone predicted that M44 is actually M45? If not, I'll claim that. (Of course it'll be somewhere around 2010 before we know I'm wrong)

Sorry Jason, that's mine already:

Originally Posted by brunoparga

My guess is that the 45th Mersenne prime to be discovered will be smaller than M44 (and M43) and will be found by a double-check by late February 2007.

(Or it might even be one of the 16 230 untested exponents below M43 / another 23 065 between M43 and M44).

Bruno

Today there's still 7208 untested exponents below "M43", plus 12113 up to "M44".

I've already lost my time prediction, but would you like to refine our smaller-than-M44 exponent prediction?

[jasong]

Originally Posted by brunoparga

Sorry Jason, that's mine already:



Today there's still 7208 untested exponents below "M43", plus 12113 up to "M44".

I've already lost my time prediction, but would you like to refine our smaller-than-M44 exponent prediction?

Since, we won't win anything by being right, I say "Yes" to sharing and refining our prediction. :) Let's go with my prediction, which only partially stepped on your toes, the idea that M44 is actually M45.