mersenne prime as a factor of another number

[Batalov]

Back on topic: Mersenne primes are factors of aliquot sequences and the conditions of their persistence in a sequence were studied. (This also is a constructed class, of course.)

[Damian]

Originally Posted by Batalov

A057468 is well studied.

It looks like is the last exponent known that makes prime. That seems to suggest that a LL-like test isn't known? (Comparing to for Mersenne primes)

[Mini-Geek]

Originally Posted by Damian

It looks like is the last exponent known that makes prime. That seems to suggest that a LL-like test isn't known? (Comparing to for Mersenne primes)

Not really. That number is 938023 bits (that's under 1M bits, whereas the largest known Mersenne prime is over 43M bits). PFGW can do a PRP test on it in roughly 73 minutes on my computer. It also says "Some of the larger entries may only correspond to probable primes." http://www.primenumbers.net/prptop/prptop.php lists it as the 24th largest PRP with 282374 digits.
It certainly took a nontrivial effort, but nothing so huge that it is proof of the use of an LL-like test.
There is a claim that "All the terms found so far are prime. [From Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Dec 18 2008]", but the 7 largest are listed at the top PRPs page, and not the prime pages. A more legitimate-looking real prime proof was claimed for everything up to 2833 at 1352 digits with Primo.

[Batalov]

Originally Posted by Mini-Geek

...There is a claim that "All the terms found so far are prime. [From Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Dec 18 2008]"

He meant exponents n, because n and not the expression values are the terms of the sequence. "found so far", in fact, should have been dropped: composite n produce algebraic factorizations with factors > 1. (For prime n, the algebraic factorization gets a trivial factor of 1.)

[CRGreathouse]

Originally Posted by Mini-Geek

There is a claim that "All the terms found so far are prime. [From Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), Dec 18 2008]", but the 7 largest are listed at the top PRPs page, and not the prime pages.

I interpret the claim differently, as "{2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827} is a subset of P".

Edit: Batalov comes to the same conclusion.

[lavalamp]

Originally Posted by Mini-Geek

A more legitimate-looking real prime proof was claimed for everything up to 2833 at 1352 digits with Primo.

Ahem, 5297 now, soon to be 7417.

When it's done I'll see about sending an update in for that sequence.

[lavalamp]

The certificate for 3^7417 - 2^7417 (3539 digits) has now been uploaded, verifications on other systems are welcome. Generated and checked on an i7 CPU running Primo 3.0.9 under Windows Vista SP2 64 bit.

http://2721.hddkillers.com/3^n-2^n/

[lorgix]

Originally Posted by lavalamp

The certificate for 3^7417 - 2^7417 (3539 digits) has now been uploaded, verifications on other systems are welcome. Generated and checked on an i7 CPU running Primo 3.0.9 under Windows Vista SP2 64 bit.

http://2721.hddkillers.com/3^n-2^n/

I verified them all on my old 32bit P4 running XP SP3.

[R.D. Silverman]

Originally Posted by Mini-Geek

Not really. That number is 938023 bits (that's under 1M bits, whereas the largest known Mersenne prime is over 43M bits). PFGW can do a PRP test on it in roughly 73 minutes on my computer. It also says "Some of the larger entries may only correspond to probable primes." http://www.primenumbers.net/prptop/prptop.php .

The claim on this page that "nobody knows how to prove or disprove its
primality" is grossly wrong. We know how to do both.

If a PRP is in fact composite, then witnesses can be found.
If it is prime, it can be proved. It may take a VERY LONG TIME,
but we certainly know how to do both computations.

[xilman]

Originally Posted by R.D. Silverman

The claim on this page that "nobody knows how to prove or disprove its
primality" is grossly wrong. We know how to do both.

If a PRP is in fact composite, then witnesses can be found.
If it is prime, it can be proved. It may take a VERY LONG TIME,
but we certainly know how to do both computations.

I seem to remember you claiming that a meet-in-the middle attack on a LL test was impossible. On that occasion I accused you of being an engineer and not a mathematician. You're now wearing your mathematician's hat.

A more charitable description of the page may be that it's incomplete (I accept that it's wrong as it stands) in that the phrase "with a reasonable amount of effort in the case that it is prime" has been omitted.


Paul

[R.D. Silverman]

Originally Posted by xilman

I seem to remember you claiming that a meet-in-the middle attack on a LL test was impossible. On that occasion I accused you of being an engineer and not a mathematician. You're now wearing your mathematician's hat.

A more charitable description of the page may be that it's incomplete (I accept that it's wrong as it stands) in that the phrase "with a reasonable amount of effort in the case that it is prime" has been omitted.


Paul

As it stands, it will be highly misleading to the less than expert reader,
because many (most?) of them will accept it as literal truth.