[QUOTE=CRGreathouse;228977]I'm not sure what you're saying. Do you have an example of a prime larger than 10^1999 which does not qualify for #19?*

* Or should I say, #19 as of 07 Sep 10 08:41 PM, since these definitions are fairly malleable.[/QUOTE]

see (2.5*10^1999)*2^2+c = 10^2000 + c so just find all c that give primes and then check for the special forms he doesn't allow.


forstep(n=1,100,2,if(isprime(10^2000+n),print(n))) is what I've been using up to you what n you search.

OK, so #19 as stated definitely covers precisely those primes greater than 10^1999. It suffices to use [TEX]c\in\{101,103\}[/TEX] and b = n = 2.

[QUOTE=CRGreathouse;229021]OK, so #19 as stated definitely covers precisely those primes greater than 10^1999. It suffices to use [TEX]c\in\{101,103\}[/TEX] and b = n = 2.[/QUOTE]

I never knew that it had a covering set and my code doesn't give 101,103 because i start at 10^2000 not 10^1999 but pretty much.

[QUOTE=science_man_88;229027]I never knew that it had a covering set and my code doesn't give 101,103 because i start at 10^2000 not 10^1999 but pretty much.[/QUOTE]

4N + 1 and 4N + 3 cover all primes > 2, right? So do that with 4N + 101 and 4N + 103.

10^1999 is the first 2000-digit number, just like 10^9 is the first 10-digit number.

I have verified the primality of 7984559573504259856359124657, a p28, through trial division.

[QUOTE=Charles]I have verified the primality of 7984559573504259856359124657, a p28, through trial division.
[/QUOTE]

I'm inclined to call bullshit, but, whatever, I won't bother. I'm busy looking for a 166350-digit prime anyway.

Odds: 1 in 7294.

So, 1 n 7294 candidates should be prime.

Each test should take about 10 minutes.

10 * 7294 = 72940 minutes ≈ 52 days.

Actually, it is 8.5 minutes. 8.5 * 7294 ≈ 43 days.

[QUOTE=3.14159;229116]I'm inclined to call bullshit, but, whatever, I won't bother.[/QUOTE]

That's your prerogative. I spent many hours trial-dividing (though, happily, my prediction on its timing was accurate -- I guessed the completion time within 5 minutes!).

Of course, your skepticism bolsters one of my points on this category: it's essentially unverifiable. By contrast, if we were using modern algorithms, we could provide certificates:
[code][2 5 1]

[3 3 1]

[11 3 1]

[103 3 1]

[149 2 1]

[9376643 3 1][/code]

Someone proposed that you could ask for residues mod all the primes less than the square root of the number, but asymptotically that takes something like [TEX]2\sqrt n/\log 2[/TEX] bits, or 32 TB in this case... using a mixed-radix method allows for smaller 'certificates', but I doubt the improvement is great.

[QUOTE=3.14159;229116]I'm busy looking for a 166350-digit prime anyway.

Odds: 1 in 7294.[/QUOTE]

If your base has no small prime factors, you must have sieved up to about 6 trillion to get odds that good.

[QUOTE=Charles]If your base has no small prime factors, you must have sieved up to about 6 trillion to get odds that good.
[/QUOTE]

I used b = 2, and indeed sieved to about 6.2 trillion.

[code][2 5 1]

[3 3 1]

[11 3 1]

[103 3 1]

[149 2 1]

[9376643 3 1][/code]

Quit using PARI, guy. It doesn't work for anything larger than 2000 digits.

[QUOTE=3.14159;229123]Quit using PARI, guy. It doesn't work for anything larger than 2000 digits.[/QUOTE]

Considering that the number has 28 digits, and 28 <= 2000, I think I was using the appropriate tool.

Basically, I'm interested in things that will expand my mind rather than simply test the speed of my CPU. When there's a good reason I'll use powerful tools outside of Pari, but for the most part if I need other tools I'm in the range of brute CPU calculations that I don't care about.