PARI's commands - Page 90

3.14159 · 2010-08-28, 04:02

Quote:

Originally Posted by CRGreathouse

How would you define this, I wonder?

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

CRGreathouse · 2010-08-28, 04:03

Quote:

Originally Posted by 3.14159

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.

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...

CRGreathouse · 2010-08-28, 04:04

Quote:

Originally Posted by 3.14159

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

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.

3.14159 · 2010-08-28, 04:05

Quote:

Originally Posted by 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...

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:

Originally Posted by 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?

I just did. See my previous post.

CRGreathouse · 2010-08-28, 04:08

Quote:

Originally Posted by 3.14159

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.

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.

3.14159 · 2010-08-28, 04:10

Quote:

Originally Posted by 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.

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.

CRGreathouse · 2010-08-28, 04:10

Quote:

Originally Posted by 3.14159

I just did. See my previous post.

[which was]

Quote:

Originally Posted by 3.14159

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

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.

3.14159 · 2010-08-28, 04:12

Quote:

Originally Posted by 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.

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

3.14159 · 2010-08-28, 04:15

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

CRGreathouse · 2010-08-28, 04:16

Quote:

Originally Posted by 3.14159

What's so ambiguous about randomly-generated numbers?

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".

3.14159 · 2010-08-28, 04:18

Quote:

Originally Posted by 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".

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.

2010-08-28, 04:15	#988
3.14159 May 2010 Prime hunting commission. 11010010000₂ Posts	Updated list: 1. Proths, where b is 2. 2. Generalized Proths, where b is any integer that is not a factorial, primorial, or prime number. 3. Factorial-based proths, where b is a factorial number. 4. Primorial-based proths, where b is a primorial number. 5. Prime-based proths, where b is a prime number. 6. Primorial, k * p(n) + 1 7. Factorial, k * n! + 1 8. Generalized Cullen/Woodall, k * b^k + 1, where b is any integer that is not a factorial, primorial, or prime number. 9. Factorial Cullen/Woodall, where b, optionally k, is a factorial number. 10. Primorial Cullen/Woodall, where b, optionally k, is a primorial number. 11. Prime-based Cullen/Woodall, where b is a prime number 12. k-b-b, numbers of the form k * b^b + 1, where b is any integer that is not a factorial, primorial, or prime number. 13. Factorial k-b-b, where b, optionally k, is a factorial number. 14. Primorial k-b-b, where b, optionally k, is a primorial number. 15. Prime-based k-b-b, where b is a prime number. 16. Number, square, and fourth, where n^1 + 1, n^2 + 1, and n^4 + 1 are all primes. 17. Special Cofactor, where the prime cofactor is of one of the forms used in this list. 18. General Cofactor, where the prime cofactor is not of a special form. 19. General arithmetic progressions, k * n + c, where c is a prime > 10^2, where the prime is at least 2000 digits in length. 20. Obsolete-tech-proven primes, 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. Last fiddled with by 3.14159 on 2010-08-28 at 04:28

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