Largest Known Prime N such that ...

[a1call]

Hey all,
What is the largest known Prime N such that nighter neither all-the-Prime-factors of N+1 nor N-1 are known?
Not looking for a probable-Prime here.

I assume the only way to prove a such a number primes would be via Trial-Factoring.

Thanks for your time.

[Dr Sardonicus]

I haven't checked that neither N-1 nor N+1 are completely factored. Assuming neither is, I can give a floor.

I just looked for largest number proved prime by some method other than those using factorizations of N-1 or N+1. And the first method I thought of was ECPP, implemented by Primo.

According to Ellipsa > Primo Top-20,

N = Partition(1289844341) (40000 decimal digits) is the largest prime certified by Primo.

[a1call]

Thank you for that.
I think there is a good chance that I am not the only one who has never heard of a partition of a number before.
If so:
https://en.m.wikipedia.org/wiki/Part...(number_theory)

Cool discovery Paul. Congrats.

[axn]

Originally Posted by a1call

Hey all,
What is the largest known Prime N such that nighter neither all-the-Prime-factors of N+1 nor N-1 are known?
Not looking for a probable-Prime here.

I assume the only way to prove a such a number primes would be via Trial-Factoring.

Thanks for your time.

To prove a prime, you don't need to know all the factors of N+1 or N-1. You only need 33% factorization. That means factordb should contain large numbers proven by N+1/N-1/combined methods which meet your criteria.

Looking at large N-1 proofs in factordb (http://www.factordb.com/nmoverview.p...age=500&skip=0) without trivial 100% N-1 factorization, I see 4469^39366+4469^19683+1 (143695 digits)

[a1call]

Yes, I forgot about that because I don't understand it.
Thanks for the refresher.

[a1call]

I assume there must be much lager ones recorded at Top-5k.
But we can leave it at that. My primary objective was to find out alternative methods to N+/-1 if any. I guess it is pretty much them and trial-factoring. ECPP is factoring regardless.

Thank you for the insights.

[sweety439]

Originally Posted by axn

To prove a prime, you don't need to know all the factors of N+1 or N-1. You only need 33% factorization. That means factordb should contain large numbers proven by N+1/N-1/combined methods which meet your criteria.

Looking at large N-1 proofs in factordb (http://www.factordb.com/nmoverview.p...age=500&skip=0) without trivial 100% N-1 factorization, I see 4469^39366+4469^19683+1 (143695 digits)

This number has trivial 50% N-1 factorization.

What is the largest number proven to be prime by N-1 / N+1 / combined methods, but without trivial factorization (i.e. N-1 / N+1 is not divisible by b^n with large n) (like the prime (791^1873-1)/790, its N-1 only has algebra factorization, and not divisible by 791^n with large n)?

[Batalov]

Originally Posted by axn

Looking at large N-1 proofs in factordb (http://www.factordb.com/nmoverview.p...age=500&skip=0) without trivial 100% N-1 factorization, I see 4469^39366+4469^19683+1 (143695 digits)

10⁴⁷⁴⁵⁰⁰ + 999 · 10²³⁷²⁴⁹ + 1 (a palindrome) has 50% factorization and is 474,501 decimal digits long.

Also,
123447^1048576-123447⁵²⁴²⁸⁸+1 is prime! (5,338,805 decimal digits) Time : 187808.030 sec.

[VBCurtis]

Originally Posted by a1call

I guess it is pretty much them and trial-factoring. ECPP is factoring regardless.

What sorts of primes do you prove by trial factoring?

In what context do you mean "ECPP is factoring regardless"?

[axn]

Originally Posted by Batalov

10⁴⁷⁴⁵⁰⁰ + 999 · 10²³⁷²⁴⁹ + 1 (a palindrome) has 50% factorization and is 474,501 decimal digits long.

Also,
123447^1048576-123447⁵²⁴²⁸⁸+1 is prime! (5,338,805 decimal digits) Time : 187808.030 sec.

Well, serves me right for not going to the primary source for primes

[axn]

Originally Posted by VBCurtis

In what context do you mean "ECPP is factoring regardless"?

In the context of "I heard Elliptic Curve and factoring is the only thing I know those are used for"

/Just-a-guess