[QUOTE=3.14159;228829]Cough, Mersenne numbers (2[sup]p[/sup] -1) are disallowed.[/QUOTE]

Sorry, I meant "Mersenne" not "Fermat". Fermat primes are of course a subset of generalized Fermat primes.

I've edited the post.

Proved the primality of 477317336992314989983, a Generalized Proth number, via trial factoring.

P.S: Can someone prove the primality of the number 29201806527798202690471270497026289647897695441591089473790387382190437432352472117744280333628938004491474620381040060030880069981 using trial factoring alone?

(Please, do not attempt. It would take you somewhere around 10[sup]60[/sup] years to do so.)

[QUOTE=3.14159;228831]p25-p27?[/QUOTE]

Low-end p25. It's been computing since before I posted about it, but then I thought that since you might disallow it I should ask.

I did a number of tests to convince myself that you'd allow it. First, I proved that the number was prime (so I won't come up with a factor toward the end of my TD). Next, I checked that neither the number plus 1 nor the number minus 1 had an unusual factorization, like Mersenne and (generalized) Fermat numbers do. Taking it further, I thought you might cry foul if you saw that it was close to a power with a large exponent (say, if my number was 2^82 - 57), so I'm now checking that there are no powers with large exponents within a million of the number.

Will that do?

Edit: Check complete. The largest exponent within a million of the number is 1, so that shouldn't be a problem.

Hmm. You would preferably use a small PRP you found, but, okay.

Also: You can use anything from the list. (Proths, k-b-b's, Cullen-Woodalls, etc.)

2[sup]82[/sup] - 57? I wouldn't be bothered. That would go under General arithmetic progressions (For the k * n - c analogues anyway.)

Here, k = 1.

All I require is that the number not be proven easily due to only having special-form potential factors.

Ex: 484550591673673379619288086628103598801942479039 = 679 * 2[sup]149[/sup] + 191 is allowed.

In general, General arithmetic progressions requires that it cannot be easily proven via N-1 testing, as Proths, k-b-b's, and Cullen-Woodalls are.

OK, proof is done!

2077756847362348863128179 is prime, and this was proven only with trial division.

[QUOTE=3.14159;228836]Hmm. You would preferably use a small PRP you found, but, okay.

Also: You can use anything from the list. (Proths, k-b-b's, Cullen-Woodalls, etc.)

2[sup]82[/sup] - 57? I wouldn't be bothered. That would go under General arithmetic progressions (For the k * n - c analogues anyway.)

Here, k = 1.

All I require is that the number not be proven easily due to only having special-form potential factors.[/QUOTE]

funny how you disallow Mersennes as k=1 n=2^x c = 1 would give Mersennes
in that category.

Okay, holding the record for TF, at a p25.

[QUOTE=science_man_88]funny how you disallow Mersennes as k=1 n=2^x c = 1 would give Mersennes
in that category[/QUOTE]

It is proven too easily. Mersennes and Fermats are disallowed.

[QUOTE=3.14159;228836]Hmm. You would preferably use a small PRP you found, but, okay.[/QUOTE]

Well, at the time that I started the test it was only a BPSW probable prime. While the test was running I did the work described (primality test, power check, etc.).

I don't know of a good way to check that the number isn't "like" 679 * 2[SUP]149[/SUP] + 191, but looking at the factorizations of the hundred numbers around mine I don't see anything funny like that. Do you?

[QUOTE=Charles]I don't know of a good way to check that the number isn't "like" 679 * 2[sup]149[/sup] + 191, but looking at the factorizations of the hundred numbers around mine I don't see anything funny like that. Do you?
[/QUOTE]

The nearest powers of 2 are 1208925819614629174706176 and 2417851639229258349412352. It's probably a general number.

Also: You can use General arithmetic progressions.

A somewhat larger example: 895941833940689770406518404950638729749847666944846720296578047412481689041272732974527851579222195722870329231843700770993569019955737882905281399 is 287 * 2[sup]480[/sup] + 887.

[QUOTE=3.14159;228841]It's probably a general number.[/QUOTE]

I don't know if it's a general number, because I don't know what a general number is. But it's not a Mersenne number and it's not a generalized Fermat number.

[QUOTE=Charles]I don't know if it's a general number, because I don't know what a general number is. But it's not a Mersenne number and it's not a generalized Fermat number.
[/QUOTE]

I'm working on a slightly larger p25, 3660797218706330586200749.