Let say I've found a very very very long prime number. I know it's prime but I need to have a proof. Is there any fast way how to check if a number is really prime?
Let say I've found the longest prime number, longer than the longest known and I want to confirm it in very short time (seconds). Is it possible?
On
If the number is of general form, fastest would be one of the modern versions of ECPP. This is several orders of magnitude faster than AKS.
If you can tolerate a tiny probability of error, the Miller-Rabin test works very well here.
If you can tolerate error, but only infinitesimally small, the Frobenius tests can provide much better worst-case error bounds than Miller-Rabin.
If the number is of special form the Lucas test or its variants might be applicable.
I guess the Fastest method known till date is - AKS Primality Test
However it won't be able to find the longest prime number within seconds.