GENERALIZED REPUNIT PROVEN PRIME

[rudy235]

The largest generalized repunit has been proven prime.

(3⁸⁷⁷⁸⁴³ - 1)/2
This number has 418838 digits and it dwarfs the second largest generalized prime repunit which has only 95202 digits. (7176²⁴⁶⁹1 - 1)/7175

It is also the most "elegant" generalized proven prime repunit with the probable exception of mersenne numbers.
As far as I know there are no Prime Generalized Repunits of base 3 in the Chris Caldwell list except (3⁴¹⁷⁷ - 1)/2

On base 10 there is (10¹⁰³¹-1)/9
On base 6 there is (6⁶⁸⁸³ - 1)/5
On base 7 there is (7¹⁶⁹⁹ - 1)/6
This are the only primes Generalized repunits published on the Chris Caldwell page (with 2<bases<11) but I suspect that at least (3³⁶⁹¹³-1) 2 with 17612 digits has to be a proven prime.
There are also other base 3 numbers that could be proven primes

(3⁴³⁰⁶³-1)/2
(3⁴⁹⁶⁸¹-1)/2
(3⁵⁷⁹¹⁷-1/2
(3⁴⁸³⁶¹¹-1)/2

[paulunderwood]

Originally Posted by rudy235

The largest generalized repunit has been proven prime.

Where is the proof?

[Jean Penné]

Hi,

jpenne@crazycomp:~$ llr64 -a10 -d grepunit.txt
Starting probable prime test of (3^877843-1)/2
Using FMA3 FFT length 72K, Pass1=384, Pass2=192, a = 3
(3^877843-1)/2 is base 3-Fermat PRP! (418838 decimal digits) Time : 600.917 sec.
Starting Lucas sequence
Using FMA3 FFT length 72K, Pass1=384, Pass2=192, P = 4, Q = 2
(3^877843-1)/2 is Fermat and Lucas PRP, Starting Frobenius test sequence
Using FMA3 FFT length 72K, Pass1=384, Pass2=192, Q = 2
(3^877843-1)/2 is Fermat, Lucas and Frobenius PRP! (P = 4, Q = 2, D = 8) Time : 3000.669 sec.
jpenne@crazycomp:~$ llr64 -a10 -d -oBPSW=1 grepunit.txt
Starting probable prime test of (3^877843-1)/2
Using FMA3 FFT length 72K, Pass1=384, Pass2=192, a = 2
(3^877843-1)/2 is base 2-Fermat PRP! (418838 decimal digits) Time : 631.337 sec.
Starting Lucas sequence
Using FMA3 FFT length 72K, Pass1=384, Pass2=192, P = 1, Q = 4
(3^877843-1)/2 is Fermat and BPSW PRP, Starting Frobenius test sequence
Using FMA3 FFT length 72K, Pass1=384, Pass2=192, Q = 4
(3^877843-1)/2 is Fermat, BPSW and Frobenius PRP! (P = 1, Q = 4, D = -15) Time : 2923.518 sec.
jpenne@crazycomp:~$

Indeed, this number is likely to be prime, but the very proof seems not to be known for now...

Regards,
Jean

[rudy235]

This is what Chris Caldwell'slist of primes says.

Note: Only proven primes are accepted on this list. These colors refer the status of this list's (re)verification process only.

The color of the (3⁸⁷⁷⁸⁴³ - 1)/2 is Probable-prime Shown to be a PRP, awaiting further testing (see note).

So there is some inconsistency there.

[paulunderwood]

Originally Posted by rudy235

This is what Chris Caldwell'slist of primes says.



The color of the (3⁸⁷⁷⁸⁴³ - 1)/2 is Probable-prime Shown to be a PRP, awaiting further testing (see note).

So there is some inconsistency there.

You are underestimating the tremendous effort that goes into proving GRUs. Finding and proving factors of N^2-1 is a daunting task. The ECPP record stands at ~35k digits. Trial division would take ... forget trial division

Incidentally, Paul Bourdelais found this PRP in 2010, long before Jinyuan Wang's recent claim.

[GP2]

The Repunit article on Wikipedia has sections on repunit probable-primes of various bases.

It provides helpful links to OEIS sequences. For exponents of base 3 repunit probable primes, this is A028491.

EDIT: wait a minute. The A028491 page states "a(18) has been proved prime by Jinyuan Wang". Didn't someone at OEIS check that before allowing it to be added?? Is there an article somewhere about the proof. Jinyuan Wang's profile page on OEIS says "I am a fan of mathematics and astronomy. I believe that mathematical discoveries do not necessarily be made by mathematicians, amateurs can also contribute to mathematics". That does not exactly inspire confidence.

It is old news that 877843 is a base-3-repunit probable prime exponent. See for instance the Lifchitz page for PRPs of this form.

If you're claiming that this has been certified prime, rather than merely probable prime... either you're a time traveler from the future, or you've discovered some deterministic algorithm similar to Lucas-Lehmer but applicable to arbitrary bases.

[rudy235]

The Repunit article on Wikipedia has sections on repunit probable-primes of various bases.

It provides helpful links to OEIS sequences. For exponents of base 3 repunit probable primes, this is A028491.

EDIT: wait a minute. The A028491 page states "a(18) has been proved prime by Jinyuan Wang". Didn't someone at OEIS check that before allowing it to be added?? Is there an article somewhere about the proof. Jinyuan Wang's profile page on OEIS says "I am a fan of mathematics and astronomy. I believe that mathematical discoveries do not necessarily be made by mathematicians, amateurs can also contribute to mathematics". That does not exactly inspire confidence.

It is old news that 877843 is a base-3-repunit probable prime exponent. See for instance the Lifchitz page for PRPs of this form.

The only thing I have no doubt is that (3^877843-1)/2 has been proven PRP a long time ago. I consider Chris Caldwell's page the standard as to prime numbers of over 400,000 digits. I believe if he is not satisfied with the proof, the number will be taken done from his page (it has happened before) and the OEIS sequence will be corrected also. No sarcasm is needed (or appreciated).

If you're claiming that this has been certified prime, rather than merely probable prime... either you're a time traveler from the future, or you've discovered some deterministic algorithm similar to Lucas-Lehmer but applicable to arbitrary bases.

[GP2]

Originally Posted by rudy235

The only thing I have no doubt is that (3^877843-1)/2 has been proven PRP a long time ago. I consider Chris Caldwell's page the standard as to prime numbers of over 400,000 digits. I believe if he is not satisfied with the proof, the number will be taken done from his page (it has happened before) and the OEIS sequence will be corrected also. No sarcasm is needed (or appreciated).

Here is Chris Caldwell's page for this number. It says the verification status is "PRP".

If a proof is claimed, the prover should have provided a primality certificate of some kind. There is no link to any certificate.

For inclusion in the sequence A028491, it's good enough that it's PRP to a sufficient number of bases. The real problem is at the bottom of that page, where it says "a(18) has been proved prime by Jinyuan Wang, Sep 07 2018". That is an unsubstantiated claim. That shouldn't have made it into the page.

Barring any unexpected mathematical breakthroughs, this exponent is far too large for primality proving. Such breakthroughs rarely come from self-proclaimed amateur non-mathematicians. It is perfectly reasonable to express skepticism.

[rudy235]

Originally Posted by GP2

Here is Chris Caldwell's page for this number. It says the verification status is "PRP".

If a proof is claimed, the prover should have provided a primality certificate of some kind. There is no link to any certificate.

For inclusion in the sequence A028491, it's good enough that it's PRP to a sufficient number of bases. The real problem is at the bottom of that page, where it says "a(18) has been proved prime by Jinyuan Wang, Sep 07 2018". That is an unsubstantiated claim. That shouldn't have made it into the page.

Barring any unexpected mathematical breakthroughs, this exponent is far too large for primality proving. Such breakthroughs rarely come from self-proclaimed amateur non-mathematicians. It is perfectly reasonable to express skepticism.

I have written to the "self-proclaimed amateur non-mathematician" asking him or her to provide a deterministic proof.

[paulunderwood]

The PRP has been removed from the top5000 primes. Jinyuan must learn to know the difference between 100% sure and (100-1/10^10^10^10^10)% sure The database's GRU top20 table has been restored.

[rudy235]

Originally Posted by paulunderwood

The PRP has been removed from the top5000 primes. Jinyuan must learn to know the difference between 100% sure and (100-1/10^10^10^10^10)% sure The database's GRU top20 table has been restored.

Yes. This ends the issue. The OEIS sequence annotation will also be restored and the number a(18) will be a considered a PRP.