A vernacular 1051 digit probable prime

[enzocreti]

3511 is the largest known Wieferich prime.




(2^3510-1 +31*73*81)/(2*31*73*729) is a spectacular 1051 digit prp!!!

[enzocreti]

Is it possible to say if it is actually prime?

[paulunderwood]

Originally Posted by enzocreti

Is it possible to say if it is actually prime?

Yes.

[enzocreti]

how?

[enzocreti]

the prime has a particular form...
2^3510-1 is divisible by 73 and 31
3511 is a Wieferich prime
81 and 729 are powers of 3

[paulunderwood]

Originally Posted by enzocreti

how?

Well, one can use Pari/GP's isprime() function given enough allocated RAM and runtime. Or use the newer ECPP capabilities of Pari/GP. Alternatively, run Primo.

[enzocreti]

Even more spectalurar:
((2^3510-1 +31*73*81)/(2*31*73*729)*9+1)/2 is also probable prime!!!

[GP2]

Originally Posted by enzocreti

Is it possible to say if it is actually prime?

Yes, it is.

For example, if you have a very recent version of PARI/GP (2.11 or higher), you can run something like:

default(parisizemax, 64000000)
cert = primecert((2^3510-1 +31*73*81)/(2*31*73*729));
s = primecertexport(cert, 1);
write("cert.out", s);

I just did this and it took only a few minutes.

Why is this PRP spectacular? It's fairly small.

[enzocreti]

((2^3510-1 +31*73*81)/(2*31*73*729)*9+1)/2is a prime congruent to 1 mod 73

[paulunderwood]

Originally Posted by enzocreti

((2^3510-1 +31*73*81)/(2*31*73*729)*9+1)/2is a prime congruent to 1 mod 73

"is a prime" -- have you proved that?

"congruent to 1 mod 73" -- so what?

[enzocreti]

it is a probable prime for factordb, moreover it is 1 mod 73 as 2^3510-1