[QUOTE=Max]A submission for #2 on the -1 side:

2778*211^47085-1 is prime![/QUOTE]

Verification, coming up:

And, this is [B]not[/B] #2.


Updated list:
1. Proths, where b is 2.
2. Generalized Proths, where b is any integer.
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
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
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 * b^n + c, where c is a prime > 10^2, where the prime is at least 2000 digits in length, and where the exponent n > 1.
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.
21. N-1 analogues of items 1-5.
22. N-1 analogues of items 6 and 7.
23. N-1 analoges of items 8-11.
24. N-1 analogues of items 12-15.
25. Obsolete-tech-proven primes, for -1 analogues only.
26. Twins.

As I said before; I will only look for 1-20.

In your case, this is item 21; N-1 analogues of items 1-5.

Max now holds the largest prime for Category 21, at 109443 digits.

Ah, I see--I was thinking of Karsten's simplified list [url=http://www.mersenneforum.org/showpost.php?p=227458&postcount=1027]here[/url], which groups the -1 analogues in the same categories as the +1. But, hey, I'm not complaining--now I get an easy largest-prime spot, rather than having to compete with Batalov's comparatively large GFN for the #2 category! :smile:

[QUOTE=3.14159;228478]Nah, that's out of the question. My skepticism isn't [B]that[/B] harsh.[/QUOTE]

I could also be a superhero who fights crime in a mask, pretending to be a computer programmer as a cover identity.

[QUOTE=3.14159;228478]Is your PARI script:

trialdivide(n) = {
forprime(p=2,(<insert primelimit here>),
if(n%p==0, return(p))
);
}
??[/QUOTE]

I said it was the straightforward trial division program, so it would have been precisely
[code]td(n)=forprime(p=2,sqrtint(n),if(n%p,return(0)));1[/code]

[QUOTE=Max]Ah, I see--I was thinking of Karsten's simplified list here, which groups the -1 analogues in the same categories as the +1. But, hey, I'm not complaining--now I get an easy largest-prime spot, rather than having to compete with Batalov's comparatively large GFN for the #2 category!
[/QUOTE]

Indeed, you do. Unless Karsten finds anything, then you're screwed.

I'm also adding a bonus category, which is technically not part of the list: Trial-division proven primes.

I will set the initial record via proving a p20.

Record set: The prime 99613292743918510357 was proven via trial division.

Update: 486226396244743118281, a p21, was proven prime in about 5 minutes using trial factoring alone.

[QUOTE=3.14159;228559]I'm also adding a bonus category, which is technically not part of the list: Trial-division proven primes.[/QUOTE]

[QUOTE=3.14159;228561]Update: 486226396244743118281, a p21, was proven prime in about 5 minutes using trial factoring alone.[/QUOTE]

I proved the primality of 618970019642690137449562111, a p27, with trial division. This was the hardest proof by trial division I had ever attempted.

I got another extention for the list:

Proving prime by trial division with pencil and paper only!

[QUOTE=Charles]I proved the primality of 590295810358705651711, a slightly larger p21, with trial division.
[/QUOTE]

You now do not hold the record..

[QUOTE=Karsten]Proving prime by trial division with pencil and paper only!
[/QUOTE]

Is the slippery slope finished?

The record for that is already there: It is 170141183460469231731687303715884105727. (Oh, wait, that is the, "by hand" record. Nevermind.)

And I'm not challenging that.

Denied category. But, the largest prime I think I have proven by hand, by trial factoring, to be prime is 6841.

[QUOTE=3.14159;228571]The record for that is already there: It is 170141183460469231731687303715884105727.
[/QUOTE]

This is a special case for trial division!

And, record snapped! The number 8905881751755136749253, a p22, was proven prime via trial division.

By hand; No one will be willing to go past 10[sup]6[/sup] or 10[sup]8[/sup]

Using PFGW, I will not be willing to go past p23-p25 with it.

I have now proved the primality of 618970019642690137449562111, a p27, with trial division. This was the hardest proof by trial division I had ever attempted.