[QUOTE=CRGreathouse]How would you define this, I wonder?
[/QUOTE]

No predictable sequences. No patterns in any section of digits.

[QUOTE=3.14159;227366]A special-form cofactor in a special-form number? Please, this only happens in Mersenne numbers. If you wanted a special-form factor, you would have to directly rig it to do so.[/QUOTE]

There are many numbers that have factors of special form, not just Mersennes. (And [i]of course[/i] I'm rigging it! Since when have I not?) But Mersenne numbers would be a great choice, probably where I'd start. You can trial-divide much more easily when you skip over 6p numbers at a stroke...

[QUOTE=3.14159;227368]No predictable sequences. No patterns in any section of digits.[/QUOTE]

Suppose this was for a competition and you needed a definition that could be programmed into a computer or written in a rulebook so there could be no disputes. How would you define it?

I'm honestly curious.

[QUOTE=CRGreathouse]There are many numbers that have factors of special form, not just Mersennes. (And of course I'm rigging it! Since when have I not?) But Mersenne numbers would be a great choice, probably where I'd start. You can trial-divide much more easily when you skip over 6p numbers at a stroke...
[/QUOTE]

Also: I don't count those because it's too easy to find a prime using those numbers. Far too easy. I require the factor to be a general prime. Or I could divide Cofactor into Special Cofactor and General Cofactor. Yeah, I'll do that.

[QUOTE=CRGreathouse]Suppose this was for a competition and you needed a definition that could be programmed into a computer or written in a rulebook so there could be no disputes. How would you define it?
[/QUOTE]

I just did. See my previous post.

[QUOTE=3.14159;227371]Also: I don't count those because it's too easy to find a prime using those numbers. Far too easy. I require the factor to be a general prime.[/QUOTE]

Why don't you re-post your list of prime records you're looking for, with these constraints and requirements listed -- no Mersenne cofactors, no small cofactors, the splitting of cofactors into 2 or 4 categories, only 'statistically likely'/un-'predictable sequences' (whatever that means), etc.

[QUOTE=CRGreathouse]Why don't you re-post your list of prime records you're looking for, with these constraints and requirements listed -- no Mersenne cofactors, no small cofactors, the splitting of cofactors into 2 or 4 categories, only 'statistically likely'/un-'predictable sequences' (whatever that means), etc.
[/QUOTE]

Split Cofactor into Special Cofactor and General Cofactor. Also: A Mersenne number for a cofactor? I never held any restriction against that. In fact, that is welcomed into Special Cofactor.

[QUOTE=3.14159;227371]I just did. See my previous post.[/QUOTE]
[which was]
[QUOTE=3.14159;227368]No predictable sequences. No patterns in any section of digits.[/QUOTE]

That's still ambiguous. I was looking for a mathematical definition, something a computer could give a firm "yes" or "no" to. I wouldn't want to search for something that I thought was fine only to have you tell me that it's not random enough for you for some reason I couldn't have guessed.

[QUOTE=CRGreathouse]That's still ambiguous. I was looking for a mathematical definition, something a computer could give a firm "yes" or "no" to. I wouldn't want to search for something that I thought was fine only to have you tell me that it's not random enough for you for some reason I couldn't have guessed.
[/QUOTE]

A computer is an idiot and is unable to understand anything. What's so ambiguous about randomly-generated numbers? (Well, at least pseudorandomly.)

Updated list:
1. [B]Proths[/B], where b is 2.
2. [B]Generalized Proths[/B], where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
3. [B]Factorial-based proths[/B], where b is a factorial number.
4. [B]Primorial-based proths[/B], where b is a primorial number.
5. [B]Prime-based proths[/B], where b is a prime number.
6. [B]Primorial[/B], k * p(n) + 1
7. [B]Factorial[/B], k * n! + 1
8. [B]Generalized Cullen/Woodall[/B], k * b^k + 1, where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
9. [B]Factorial Cullen/Woodall[/B], where b, optionally k, is a factorial number.
10. [B]Primorial Cullen/Woodall[/B], where b, optionally k, is a primorial number.
11. [B]Prime-based Cullen/Woodall[/B], where b is a prime number
12. [B]k-b-b[/B], numbers of the form k * b^b + 1, where b is any integer that is [B]not[/B] a factorial, primorial, or prime number.
13. [B]Factorial k-b-b[/B], where b, optionally k, is a factorial number.
14. [B]Primorial k-b-b[/B], where b, optionally k, is a primorial number.
15. [B]Prime-based k-b-b[/B], where b is a prime number.
16. [B]Number, square, and fourth[/B], where n^1 + 1, n^2 + 1, and n^4 + 1 are all primes.
17. [B]Special Cofactor[/B], where the prime cofactor is of one of the forms used in this list.
18. [B]General Cofactor[/B], where the prime cofactor is not of a special form.
19. [B]General arithmetic progressions[/B], k * n + c, where c is a prime > 10^2, where the prime is at least 2000 digits in length.
20. [B]Obsolete-tech-proven primes[/B], using the original PrimeForm or Proth.exe, or any other prime to prove primality of any type of prime listed here. Note: The prime must be at least 7500 digits in length.

Oh: Mersennes are not in the list. Ah, well. Mersenne number cofactors are classified as General Cofactor, then.

[QUOTE=3.14159;227375]What's so ambiguous about randomly-generated numbers?[/QUOTE]

Because if I hand you a number, you don't know where it came from. At this point I'm not even sure what you intend, let alone how precisely to define it. You've described it differently each time: "statistically likely", "No predictable sequences. No patterns in any section of digits.", and now "randomly-generated".

[QUOTE=CRGreathouse]Because if I hand you a number, you don't know where it came from. At this point I'm not even sure what you intend, let alone how precisely to define it. You've described it differently each time: "statistically likely", "No predictable sequences. No patterns in any section of digits.", and now "randomly-generated".
[/QUOTE]

No patterns in any section of digits.
No predictable sequences.
Statistically likely.

Don't those stand out to you as the characteristics of a randomly-chosen number?

Also: View above, I posted the updated list.