Re: [Cfrg] Prime 630*(427!+1)+1 for classic DH?

Greg Rose <ggr@seer-grog.net> Wed, 05 April 2017 23:20 UTC

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From: Greg Rose <ggr@seer-grog.net>
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Date: Wed, 5 Apr 2017 16:20:26 -0700
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Subject: Re: [Cfrg] Prime 630*(427!+1)+1 for classic DH?
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On Apr 5, 2017, at 13:32 , Greg Rose <ggr@seer-grog.net> wrote:
> 
> 
>> On Apr 5, 2017, at 12:39 , Dan Brown <danibrown@blackberry.com> wrote:
>> [snip]... for the choice 427!+1 of large prime factor of p-1. [snip]
>> 
> 
> Since a surprisingly large number of people think that N!+1 must be prime (referring back to the proof that there are an infinite number of primes) I checked. It is. Not that I ever doubted you, Dan...

A number of people responded off-list asking why I said this, and/or how I checked primality.

Quoting: https://primes.utm.edu/notes/proofs/infinite/euclids.html
> Theorem.
> There are infinitely many primes.
> Proof.
> Suppose that p1=2 < p2 = 3 < ... < pr are all of the primes. Let P = p1p2...pr+1 and let p be a prime dividing P; then p can not be any of p1, p2, ..., pr, otherwise p would divide the difference P-p1p2...pr=1, which is impossible. So this prime p is still another prime, and p1, p2, ..., pr would not be all of the primes.
> It is a common mistake to think that this proof says the product p1p2...pr+1 is prime.  The proof actually only uses the fact that there is a prime dividing this product.

I calculated 427!+1 using bc, edited it to remove continuation lines, and pasted the result into https://www.alpertron.com.ar/ECM.HTM. If it had been composite, it might have taken a long time when asked to factor it, but it takes very little time to come back and say that it's prime (that is, it is its only non-trivial factor).

However, I then proceeded with the obvious experiment to check the original number: (quoting Dan's original mail)
> Is the prime p=630*(427!+1)+1 vulnerable to the SNFS, or some variant of SNFS?  I think not, but I could easily be very wrong.

Somewhat to my dismay, it did NOT immediately pronounce the number to be prime. 20 minutes later it's still trying to factor it. I don't have anything to hand that will just check primality... perhaps someone else should check it. (Or I will try harder when I have something better than my laptop, which will be a while). It's also possible I made some sort of transcription error, and if so I apologize. Someone who has Mathematica or Maple lying around?

Greg.