Double check - this is prime, yes?

[CRGreathouse]

Double check -- this is prime, yes?

I thought I had uncovered some kind of pseudoprime, since I found this number with pfgw and on checking it, found it to be composite. I tested it for pseudoprimality with a battery of tests, though, and it passed all of them -- leading me to think that I was mistaken in the first place and tested some other number by typo. But on the off chance it is a pseudoprime I'd like to know, since it would be the only known Baillie-Pomerance-Selfridge-Wagstaff pseudoprime.

The 9928-digit number is:
32986!/(2*(16493!)^2)+1

 

[CompuChip]

After about 4 minutes of thinking very hard, Mathematica confirms that
32986!/(2*(16493!)^2) + 1
is prime.

PS It found no divisors but 1 and itself. Which could be viewed as a confirmation of either the fact that it's prime, or that the PrimeQ[] function first finds the divisors and then checks how many there are

 

[CRGreathouse]

I don't imagine you know what kind of test it used to determine that? I've run a (B)PSW test (a 2-strong plus a (21, -1)-Lucas test in this case) and 10 strong (Rabin-Miller) tests with random bases. The latter has worst-case error of 2^-20.

 

[rocomath]

how much math should i have before i attempt to self-study number theory?

 

[Kummer]

how much math should i have before i attempt to self-study number theory?

It depends what kind. The most basic concepts form Elementary Number Theory and can even be learned while still in high school.

 

[CompuChip]

I don't imagine you know what kind of test it used to determine that?

The http://reference.wolfram.com/mathematica/note/SomeNotesOnInternalImplementation.html#6849 say that

• PrimeQ first tests for divisibility using small primes, then uses the Miller-Rabin strong pseudoprime test base 2 and base 3, and then uses a Lucas test.
• As of 1997, this procedure is known to be correct only for n<1016, and it is conceivable that for larger n it could claim a composite number to be prime.

I'm now re-running it using this package.
[update]The kernel crashed and I need it for something else now. It ran for over an hour though. [/update]

 

[CRGreathouse]

The http://reference.wolfram.com/mathematica/note/SomeNotesOnInternalImplementation.html#6849 say that

• PrimeQ first tests for divisibility using small primes, then uses the Miller-Rabin strong pseudoprime test base 2 and base 3, and then uses a Lucas test.
• As of 1997, this procedure is known to be correct only for n<1016, and it is conceivable that for larger n it could claim a composite number to be prime.

I'm now re-running it using this package.
[update]The kernel crashed and I need it for something else now. It ran for over an hour though. [/update]

Thanks, that looks good.

I think the number is too long to reasonably prove prime, but it looks like we have shown it to be very likely to be a prime.