[QUOTE=3.14159;228816]Any random prime integer.[/QUOTE]

OK, but what's random and how can you tell?

In particular, I'd like to make a submission and I'd like to know if you'll allow it before spending too much effort. (A large majority of the times I've submitted something for inclusion on one of your lists, you've changed the rules to disallow it.)

[QUOTE=3.14159;228816]I disallow: Mersennes and Fermats, because they have special-form factors.[/QUOTE]

My number is not of the form 2^n - 1, nor is it of the form 2^(2^n) + 1.

[QUOTE=Charles]In particular, I'd like to make a submission and I'd like to know if you'll allow it before spending too much effort. (A large majority of the times I've submitted something for inclusion on one of your lists, you've changed the rules to disallow it.)
[/QUOTE]

Because you use trickery! You always use trickery!

Also: For which item would you like to make a submission?

[QUOTE=Charles]My number is not of the form 2^n - 1, nor is it of the form 2^(2^n) + 1.
[/QUOTE]

And it should not be of the form b[sup]2[sup]n[/sup][/sup] + 1 (Generalized Fermat), either.

[QUOTE=Charles]OK, but what's random and how can you tell?
[/QUOTE]

Step 1. Pick a number.
Step 2. Trial division up to its square root.
Step 3. If prime, report here.

That is, if you are applying for the unofficial trial factoring proof.

[QUOTE=3.14159;228819]For which item would you like to make a submission?[/QUOTE]

Trial-division prime.

[QUOTE=3.14159;228819]And it should not be of the form b[sup]2[sup]n[/sup][/sup] + 1 (Generalized Fermat), either.[/QUOTE]

It's not of that form. Is that good enough?

[QUOTE=Charles]Trial-division prime.
[/QUOTE]

Excellent.

[QUOTE=Charles]It's not of that form. Is that good enough?
[/QUOTE]

I still think you have one last piece of trickery up your sleeves..

[QUOTE=3.14159;228820]Step 1. Pick a number.
Step 2. Trial division up to its square root.
Step 3. If prime, report here.[/QUOTE]

I've done step 1, but I'm not sure if you'll accept it. You've denied almost all of my previous attempts.

I'm working on #2. The number is large (though of course not as large as my Mersenne) so this step is hard.

[QUOTE=Charles]I've done step 1, but I'm not sure if you'll accept it. You've denied almost all of my previous attempts.
[/QUOTE]

They were Mersenne numbers. Trickery!

[QUOTE=Charles]I'm working on #2. The number is large (though of course not as large as my Mersenne) so this step is hard.
[/QUOTE]

I'm guessing it's a p22-p24?

Mersenne primes and generalized Fermat primes are disallowed, but other numbers are good (provided, of course, that you actually do the trial division up to the square root).

The largest number I have tried thus far is 8905881751755136749253, a general p22.

[QUOTE=Charles]Fermat primes and generalized Fermat primes are disallowed, but other numbers are good (provided, of course, that you actually do the trial division up to the square root).
[/QUOTE]

Cough, Mersenne numbers (2[sup]p[/sup] -1) are disallowed.

[QUOTE=3.14159;228826]They were Mersenne numbers. Trickery![/QUOTE]

Mersenne numbers were allowed at the time that I posted them.

[QUOTE=3.14159;228826]I'm guessing it's a p22-p24?[/QUOTE]

No. It's either <= 9999999999999999999973 or >= 100000000000000000000117.

[QUOTE=Charles]No.
[/QUOTE]

p25-p27?