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 2020-11-27, 01:14 #1 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 36758 Posts Largest Known Prime N such that ... 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. Last fiddled with by Dr Sardonicus on 2020-11-27 at 01:20 Reason: Attempt at gixnif optsy
 2020-11-27, 02:43 #2 Dr Sardonicus     Feb 2017 Nowhere 2×2,087 Posts 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.
 2020-11-27, 02:57 #3 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7BD16 Posts 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. Last fiddled with by a1call on 2020-11-27 at 02:58
2020-11-27, 03:29   #4
axn

Jun 2003

3·5·17·19 Posts

Quote:
 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)

 2020-11-27, 03:34 #5 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 36758 Posts Yes, I forgot about that because I don't understand it. Thanks for the refresher.
 2020-11-27, 04:51 #6 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 7×283 Posts 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.
2020-11-27, 05:57   #7
sweety439

Nov 2016

22×691 Posts

Quote:
 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)?

Last fiddled with by sweety439 on 2020-11-27 at 05:58

2020-11-27, 06:46   #8
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

2×11×421 Posts

Quote:
 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,
1234471048576-123447524288+1 is prime! (5,338,805 decimal digits) Time : 187808.030 sec.

2020-11-27, 07:21   #9
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

35·19 Posts

Quote:
 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"?

2020-11-27, 11:45   #10
axn

Jun 2003

3·5·17·19 Posts

Quote:
 Originally Posted by Batalov 10474500 + 999 · 10237249 + 1 (a palindrome) has 50% factorization and is 474,501 decimal digits long. Also, 1234471048576-123447524288+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

2020-11-27, 11:47   #11
axn

Jun 2003

484510 Posts

Quote:
 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

Last fiddled with by axn on 2020-11-27 at 11:49

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