[QUOTE=science_man_88;231149]you say I'm solving for t without proof of that and my statement of the opposite yet you still claim it hence you assume it's true there's the assumption.[/QUOTE]

I posted a problem for you which, with the variables you're using, requires you to solve for t. You don't have to solve for t -- you could solve for p, or play a game of baseball, or whatever -- but in that case you wouldn't be doing the proposed problem.

I don't see a problem asking for t at all actually but do as you want.

[QUOTE=science_man_88;231161]I don't see a problem asking for t at all actually but do as you want.[/QUOTE]

Uh... okay?

To clarify, if that's needed: insofar as a person is solving the problem I posed in #618 with your equation, they would solve for t.

[QUOTE=CRGreathouse;230984]I was looking for a function that, given a probability p (and N, t, L), would give an n such that

estimatePrimes(N,t,n,L)

is 1 - p.[/QUOTE]

then this post is flawed as are all after it.

How would one of you go about proving or disproving the primality of 3379212930002668486657 using only hand calculations? It has no factors under 16777216.

[code](19:41) gp > Mod(13, 3379212930002668486657)^1689606465001334243328
%73 = Mod(1859858123868907490075, 3379212930002668486657)[/code]

Nevermind, the number is composite.

3379212930002668486657 = 30245153 * 111727420588769

Okay, 3773512084578210152449 is at least a PRP. How would you go about in proving it prime, with no computer assistance?

I've heard theres a way with subtracting multiples of the divisor you want to check until you either prove or disprove the resultant.

@Sm88: Too impractical to do a few million times.

The shortest idea I have so far is manually using a few M-R tests.

Or applying Proth's theorem by hand, as the number in question (3773512084578210152449) is a Proth number.

27009 * 10^20 + 1 = 7^2 * 55120408163265306122449;

55120408163265306122449 is a twin prime.

[QUOTE=3.14159;231176]The shortest idea I have so far is manually using a few M-R tests.[/QUOTE]

I'm not sure how you could use this, by hand, to prove primality. Actually, considering that it's 72 bits, I'm not even sure of how I could use that to prove primality with a computer.

[QUOTE=3.14159;231176]Or applying Proth's theorem by hand, as the number in question (3773512084578210152449) is a Proth number.[/QUOTE]

That's probably the most reasonable way. It would be a pain, though.

[QUOTE=Charles]I'm not sure how you could use this, by hand, to prove primality. Actually, considering that it's 72 bits, I'm not even sure of how I could use that to prove primality with a computer.
[/QUOTE]

About 20-25 tests prove that it is prime. Oh, right.. M-R's weakness = Proth numbers.

You can prove any non-Proth 72-bit number prime with M-R.

Ex: [code](21:29) gp > ispseudoprime(8183202143983816686553, 30)
%130 = 1[/code]

That was a 72-bit number.

[QUOTE=Charles]That's probably the most reasonable way. It would be a pain, though.
[/QUOTE]

It depends on a. Suppose a = 23;

23^1886756042289105076224 modulo 3773512084578210152449?

Multiplication of 23 by itself has to happen at least 16 times for it to be ≥ 3773512084578210152449.

Also, a good idea would be to identify when the mod results repeat. (If they do repeat.)

Ex: Powers of 71, mod 23: 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 9, 18, ...

6^134217728 + 1 is divisible by 51808043009.

Two factors for 5^36893488147419103232 + 1: 221360928884514619393 and 2434970217729660813313 both divide 5^36893488147419103232 + 1.

Probably well-known at the moment;

I betcha no one can find the smallest divisor of 10000^(2^160) + 1, which = 10^(2^162) + 1.