Thread for posting tiny primes

[3.14159]

Originally Posted by Max

A submission for #2 on the -1 side:

2778*211^47085-1 is prime!

Verification, coming up:

And, this is not #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.

[mdettweiler]

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!

[CRGreathouse]

Originally Posted by 3.14159

Nah, that's out of the question. My skepticism isn't that harsh.

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

Originally Posted by 3.14159

Is your PARI script:

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

I said it was the straightforward trial division program, so it would have been precisely

td(n)=forprime(p=2,sqrtint(n),if(n%p,return(0)));1

[3.14159]

Originally Posted by 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!

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.

[3.14159]

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

[CRGreathouse]

Originally Posted by 3.14159

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

Originally Posted by 3.14159

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

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

[kar_bon]

I got another extention for the list:

Proving prime by trial division with pencil and paper only!

[3.14159]

Originally Posted by Charles

I proved the primality of 590295810358705651711, a slightly larger p21, with trial division.

You now do not hold the record..

Originally Posted by Karsten

Proving prime by trial division with pencil and paper only!

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.

[kar_bon]

Originally Posted by 3.14159

The record for that is already there: It is 170141183460469231731687303715884105727.

This is a special case for trial division!

[3.14159]

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

By hand; No one will be willing to go past 10⁶ or 10⁸

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

[CRGreathouse]

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.