just an informal question
 

I know exponents past 1 billion have been tested..

So what if any is the actual testable limit at this time,
and also what is the effective limit if smaller?

How many exponents over 100 million have been tested?

Is there a locus of primality likeliness being searched?
Do you have an expectation of where the next MP lies?

Just consider these informal questions not requiring
detailed answers.

It seems like some of these questions have answers at [url]https://www.mersenne.org/primenet/[/url]

[url]http://www.mersenneforum.org/showthread.php?p=463062#post463062[/url] (You should read most of that thread).

[QUOTE=davar55;464142]Is there a locus of primality likeliness being searched?
Do you have an expectation of where the next MP lies?[/QUOTE]
It is a blatant fallacy to search for a random event where its median expectation (or mode of expectation distribution or whatever similar, like 'a locus of primality likeliness') is located.

Take a Geiger counter and a very weak particle emitter.
Let's say that we know that in the next 24 hours you will hear 24 clicks (because over the last 2 days that held true). Show me a person who will go out of the room for a long coffee break and will come back exactly for one minute exactly after 1 hour to listen for the click - and I will show you a fool.

[QUOTE=Batalov;464152]It is a blatant fallacy to search for a random event where its median expectation (or mode of expectation distribution or whatever similar, like 'a locus of primality likeliness') is located.

Take a Geiger counter and a very weak particle emitter.
Let's say that we know that in the next 24 hours you will hear 24 clicks (because over the last 2 days that held true). Show me a person who will go out of the room for a long coffee break and will come back exactly for one minute exactly after 1 hour to listen for the click - and I will show you a fool.[/QUOTE]

:goodposting:

[QUOTE=Batalov;464152]Show me a person who will go out of the room for a long coffee break and will come back exactly for one minute exactly after 1 hour to listen for the click - and I will show you a fool.[/QUOTE]

Trump.

we know they can't be in cunningham chains of the first kind ... ( except the last member, edit: or the first one if it's 1 mod 4, or if it's 3)

[QUOTE=Batalov;464152]Take a Geiger counter and a very weak particle emitter.
Let's say that we know that in the next 24 hours you will hear 24 clicks (because over the last 2 days that held true). Show me a person who will go out of the room for a long coffee break and will come back exactly for one minute exactly after 1 hour to listen for the click - and I will show you a fool.[/QUOTE]

My potassium chloride (bottle of NoSalt) is like that. I have to take a five minute count to detect its gamma emissions over background.

[QUOTE=Batalov;464152]It is a blatant fallacy to search for a random event where its median expectation (or mode of expectation distribution or whatever similar, like 'a locus of primality likeliness') is located.
...
[/QUOTE]

OK. I was just referring to I thought I had noticed activity around exponents
near 75000000 before the 60xxxxxx's had been explored, and didn't want to
attribute your finding M49 to fortune.

[QUOTE=davar55;464199]OK. I was just referring to I thought I had noticed activity around exponents
near 75000000 before the 60xxxxxx's had been explored, and didn't want to
attribute your finding M49 to fortune.[/QUOTE]

There are no known patterns of distribution of Mersenne primes.

At one time there was a purely empirical "islands" conjecture, inspired by the fact that 1257787 is close to 1398269, and 2976221 is close to 3021377. But this seems to be discredited now.

However...

All Mersenne prime exponents [I]p[/I] are themselves prime, and therefore either [I]p[/I] ≡ 1 (mod 4) or [I]p[/I] ≡ 3 (mod 4), except for the trivial case of [I]p[/I] = 2. According to the [URL="http://primes.utm.edu/mersenne/heuristic.html"]Lenstra–Pomerance–Wagstaff conjecture[/URL], the two cases (1 and 3) do [I]not[/I] produce Mersenne primes with exactly equal probability.

See the line numbered "3." in the above link. If I'm doing the math correctly, for [I]p[/I] in the current ranges for double-checking and first-time checking, [I]p[/I] ≡ 1 (mod 4) would be about 6% more likely to produce a Mersenne prime than [I]p[/I] ≡ 3 (mod 4).

This isn't enough to make it worthwhile to narrow the search candidates.

The disparity is larger for small [I]p[/I], for instance it would amount to 11% more likely for [I]p[/I] around 10000. Since most of the 49 known Mersenne primes are for relatively small [I]p[/I], it's perhaps not a surprise that 29 of them have [I]p[/I] ≡ 1 (mod 4), 19 of them have [I]p[/I] ≡ 3 (mod 4), and the remaining one is of course [I]p[/I] = 2.

PS,
It does seem that we have been uncommonly fortunate with recent Mersenne prime finds. There are eleven of them between [I]p[/I] = 10[SUP]7[/SUP] and 10[SUP]8[/SUP], and we haven't even tested the whole range yet. For other ranges we have 10[SUP]1[/SUP]–10[SUP]2[/SUP]: six, 10[SUP]2[/SUP]–10[SUP]3[/SUP]: four, 10[SUP]3[/SUP]–10[SUP]4[/SUP]: eight, 10[SUP]4[/SUP]–10[SUP]5[/SUP]: six, 10[SUP]5[/SUP]–10[SUP]6[/SUP]: five, 10[SUP]6[/SUP]–10[SUP]7[/SUP]: five.

Note that there was a factor of more than 4.1 between the exponents 127 and 521, which turned out to be the gap between the pre-computer era and the computer era. Spare a thought for manual calculators like Horace S. Uhler, who diligently worked that gap in vain. If by extremely unlucky chance there was a comparable multiple between the current record Mersenne prime exponent 74207281 and the next highest Mersenne prime exponent, then the latter would not be before [I]p[/I] = 300 million (!) In that case maybe the historical record will show a similar gap between the conventional-computer era and the quantum-computing era.

[QUOTE=GP2;464207]
Note that there was a factor of more than 4.1 between the exponents 127 and 521, which turned out to be the gap between the pre-computer era and the computer era. Spare a thought for manual calculators like Horace S. Uhler, who diligently worked that gap in vain. If by extremely unlucky chance there was a comparable multiple between the current record Mersenne prime exponent 74207281 and the next highest Mersenne prime exponent, then the latter would not be before [I]p[/I] = 300 million (!) In that case maybe the historical record would show a similar gap between the conventional-computer era and the quantum-computing era.[/QUOTE]

:goodposting:

Surely the gods would not be so cruel...

BTW what is the ETA for regularly testing exponents near 300M?