[QUOTE=Dobri;583436]... it is about devising strategies (if at all possible) for manual testing.[/QUOTE]
[I]1 Corinthians 13:11[/I]
"When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things."

[QUOTE=Dobri;583437]But let me propose a friendly bet: If the digit sum of the next discovered Mersenne prime exponent is less than or equal to 41, then the forum will reopen the closed thread. And if the digit sum is greater than 41, I will admit the fallacy of my reasoning and leave the forum.[/QUOTE]That isn't how mathematics works.

You either have some mathematical basis or you don't. No amount of bets or empirical results will change that.

[QUOTE=Batalov;583439][I]1 Corinthians 13:11[/I]
"When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things."[/QUOTE]
That's good for you. I would add to that also to keep an open mind.

[QUOTE=retina;583440]That isn't how mathematics works.

You either have some mathematical basis or you don't. No amount of bets or empirical results will change that.[/QUOTE]
As I wrote in another post, this is an empirical work at this stage. The mathematical basis has to be extended to cover the gray area after trials and errors (lots of them).

I will at some point implement a method so my dog picks an exponent! Getting her to tread on the keyboard should be easy enough.

I did at one point test a few exponents which have a twin-prime below the exponent I was testing, based on this observation
[URL]https://www.mersenneforum.org/showthread.php?t=26812[/URL]
However, manual assignments are larger than the category 0 & 1 exponents I can get from the server, so take longer to test. I guess, if I wanted, I could set [FONT=Arial]GetMinExponent [/FONT]& [FONT=Arial]GetMaxExponent [/FONT]in local.txt to get category 1 assignments of the twin-primes. But I'm not convinced the observation is any more than pure chance, so I am not bothering.

There's someone on here doing testing of exponents in a certain range, based on some mad idea he has. I think I would keep quite if I had such a stupid idea, but otherwise I don't see any reason to keep quite about testing strategies.

[QUOTE=Dobri;583442]As I wrote in another post, this is an empirical work at this stage. The mathematical basis has to be extended to cover the gray area after trials and errors (lots of them).[/QUOTE]So far you have no maths basis, so it isn't extending, it is creating.

And so far no one has said you are wrong (or right). You might be right, but without some actual analysis other than "I eyeballed the graph" then there is no way to know.

And making your claim contingent upon a single future result is very reckless IMO. Your claim might still be correct (we don't know) even if the next prime is outside of what you predict.

[QUOTE=drkirkby;583443]I will at some point implement a method so my dog picks an exponent! Getting her to tread on the keyboard should be easy enough.

I did at one point test a few exponents which have a twin-prime below the exponent I was testing, based on this observation
[URL]https://www.mersenneforum.org/showthread.php?t=26812[/URL]
However, manual assignments are larger than the category 0 & 1 exponents I can get from the server, so take longer to test. I guess, if I wanted, I could set [FONT=Arial]GetMinExponent [/FONT]& [FONT=Arial]GetMaxExponent [/FONT]in local.txt to get category 1 assignments of the twin-primes. But I'm not convinced the observation is any more than pure chance, so I am not bothering.

There's someone on here doing testing of exponents in a certain range, based on some mad idea he has. I think I would keep quite if I had such a stupid idea, but otherwise I don't see any reason to keep quite about testing strategies.[/QUOTE]
Your dog would serve as an excellent assistant to randomly select exponents.
Manual testing is concerned with Category 4 assignments and they take much longer to test indeed (years per exponent).
Anyway, using empirical strategies or not, the amount of work done by the volunteers would remain the same, so the GIMPS performance would remain unaffected.

[QUOTE=retina;583444]So far you have no maths basis, so it isn't extending, it is creating.
And making your claim contingent upon a single future result is very reckless IMO. Your claim might still be correct (we don't know) even if the next prime is outside of what you predict.[/QUOTE]
It is a calculated risk. I do not expect a new Mersenne prime to be found in the next 3-5 years.
Meanwhile, substantial work could be done to better understand the empirical observations.

[QUOTE=Dobri;583436]...
The observations can be simple (even elementary) but there are no meaningless observations as soon or later they serve their purpose to connect the dots.
...[/QUOTE]
That assumes every observation gets us closer to some grand theorem, the vast majority of observations actually obscure what mathematical dots can be meaningfully connected. Don't get me wrong it can be fun to be a crank and massage sequences and concepts into some eldritch horror of a theorem, but in the end it's no more meaningful than completing a crossword puzzle.

[QUOTE=Dobri;583404]Finally, there was the post by a user who attached a fake graph (with non-normalized probabilities exceeding unity, and using a linear heuristic they admitted in a preceding post to not necessarily believe in) to 'compare' it with the tiny sample size of known Mersenne primes with the intention to disprove the OP (actually, the upper part of the fake graph tends to zero, which is in accordance with the zero Mersenne primes discovered in that upper domain) and asking for the thread to be closed.[/QUOTE]

:soapbox:

In the words of Joe Biden, "C'mon, man!"

Before you immediately accuse someone of malice, stop and think whether there might be any other explanation for what they've done.

It's true that for p=3 and p=5, the heuristic gives a probability exceeding 1. But the fact that I didn't adjust these probabilities down to exactly 1 (after all, we know they're prime) in the graph was an honest mistake; I just hadn't checked whether the heuristic always gave reasonable values for small primes. I only discovered the mistake later, and I didn't correct the graph because I hoped any sane person would understand that this would only matter for a couple of very small values of p. I've attached a corrected version so you can see just how little difference there is. :deadhorse:

It's not that I don't necessarily believe in the heuristic. There are good reasons for thinking it might be true, and I have no reason to disbelieve it, but I would not be totally shocked if someone were to prove it wrong, whereas I would be shocked if someone disproved the Goldbach conjecture, for example. By the way, the heuristic is only linear in the limit. The version I've been using is the following:

[quote]The probability that 2^p-1 is prime is about (e^gamma log ap )/(p log 2) where a=2 if p=3 (mod 4), and a=6 if p=1 (mod 4).[/quote]

The linear fit comes from approximating log ap as log p, which then cancels with the 1/(log p) probability that p is prime, making the integral easier. I haven't been making this approximation.

Of course the probabilities for higher digit sums tend to zero, because we've only searched up to p a little over 100M. That doesn't contradict any of what I said. The graph does not say anything about probabilities for higher exponents that we haven't searched yet!

[QUOTE=M344587487;583449]That assumes every observation gets us closer to some grand theorem, the vast majority of observations actually obscure what mathematical dots can be meaningfully connected. Don't get me wrong it can be fun to be a crank and massage sequences and concepts into some eldritch horror of a theorem, but in the end it's no more meaningful than completing a crossword puzzle.[/QUOTE]
This may still sound ridiculous in number theory but computational advances in pattern analysis become increasingly powerful.

Rather than proving a grand theorem, one could potentially use software tools to eliminate possibilities and end up with a limited set of exponents for a given strategy. Please do not misunderstand me, we are far from applying this directly in number theory at present.

But if thinking about it strategically, imagine a future version of Prime95 in the next 10-15 years (I hope to be still around) which has "Select Assignment" and also "Select Team Strategy" options.