GENERALIZED REPUNIT PROVEN PRIME

[MisterBitcoin]

Originally Posted by paulunderwood

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.

I tryed to prove an 20k dd PRP from factordb, after 3 months (!) I´ve passed about 10% from phase 1 ECCP.

It was only 20k dd, now take a look at the PRP Top page...
It wont be possible to prove such a number rudy guessed it´s a proven prime, unless you work for the NSA and have an console version of primo...

[paulunderwood]

Originally Posted by MisterBitcoin

I tryed to prove an 20k dd PRP from factordb, after 3 months (!) I´ve passed about 10% from phase 1 ECCP.

It was only 20k dd, now take a look at the PRP Top page...
It wont be possible to prove such a number rudy guessed it´s a proven prime, unless you work for the NSA and have an console version of primo...

Hopefully one day there will be a net version of Primo, so we can all work on things like the repunit R49081.

[wpolly]

Originally Posted by paulunderwood

Hopefully one day there will be a net version of Primo, so we can all work on things like the repunit R49081.

Considering R49081-1 has a ~17% factored part, would an hybrid APR-CL test be more efficient in this case?

[rudy235]

Originally Posted by wpolly

Considering R49081-1 has a ~17% factored part, would an hybrid APR-CL test be more efficient in this case?

A CHG Proof would require at least 25+% of factorization of N-1

We are not close to that. Even the factorization of the smallest composite factor of R49081the 320 digit composite equivalent to (1878270012......) has proven to be very challenging


Φ409(10)= 1637 × 13907 × 77711 × 1375877 × 2777111 × 5371851809 x C16 X C39 X C320

[sweety439]

See the project https://mersenneforum.org/showthread.php?t=21808 for the searching for the smallest generalized repunit (probable) prime for bases 2<=b<=1024 and -1024<=b<=-2.

[paulunderwood]

Let Rp be a p-digit repunit. And d=(Rp-1)/p. Apart from R2, the next 3 repunit primes meet the condition lift(Mod(9,Rp)^(d%p))%p==d%p+2. Unfortunately, the next known primes/PRPs do not, although some composites do -- R2511, R73783 and R364759. I have checked all repunits up to 24 million digits long with no factors less than 10000*2*p+1 and found no others meeting the above criterion.

[rudy235]

Originally Posted by paulunderwood

Let Rp be a p-digit repunit. And d=(Rp-1)/p. Apart from R2, the next 3 repunit primes meet the condition lift(Mod(9,Rp)^(d%p))%p==d%p+2. Unfortunately, the next known primes/PRPs do not, although some composites do -- R2511, R73783 and R364759. I have checked all repunits up to 24 million digits long with no factors less than 10000*2*p+1 and found no others meeting the above criterion.

Let me understand. You are saying that you have checked Repunits up to 24,000,000 and that you have not found any PRIMES/PRP or composites that follow that criterion.

But that (if I understand correctly) does not mean that —besides the 9 primes/PRP A004023 that are already known— there are no other PRP's up to 24,000,000.

[paulunderwood]

Originally Posted by rudy235

Let me understand. You are saying that you have checked Repunits up to 24,000,000 and that you have not found any PRIMES/PRP or composites that follow that criterion.

But that (if I understand correctly) does not mean that —besides the 9 primes/PRP A004023 that are already known— there are no other PRP's up to 24,000,000.

The criterion is "flaky". There could be new repunit PRPs with less than 24 million digits. If you are interested in searching for them there is an excellent website coordinating the search for them.

[rudy235]

Prime repunits are listed by the number of digits in exponent n

1 digit A2 =2 
2 digits A3 and A4 19, 23
3 digits A4 317
4 digits A5 1031
5 digits A6 49081 and A7 86453
6 digits A8 109297 and A9 270343
7 digits (up to 3657959) no PRPs

If we divide the exponents we see that the largest "gap" logarithmically is between A(4) and A(5) . log(49081/1031) ~ 1.67 . where 10^1.67 ~47.6

For the 9 prime repunits the average gap is log(2700343/2)/8 ~0.64 where 10^0.64 ~5,84
But between A(9) and a potential A(10) there is —at the very least— a logarithmic gap of 1.13

Between mersenne primes (a generalized repunit base 2) the largest logarithmic gap is between A₁₂ and A₁₃ and is 0.61 . The theoretical mean ratio is 2^(1/e^∂) or about 1.48 . Does anyone know what is the theoretical ratio between prime repunits?

Seems there is scarcity of repunits compared to other generalized repunits.

[sweety439]

Originally Posted by rudy235

Prime repunits are listed by the number of digits in exponent n

1 digit A2 =2 
2 digits A3 and A4 19, 23
3 digits A4 317
4 digits A5 1031
5 digits A6 49081 and A7 86453
6 digits A8 109297 and A9 270343
7 digits (up to 3657959) no PRPs

If we divide the exponents we see that the largest "gap" logarithmically is between A(4) and A(5) . log(49081/1031) ~ 1.67 . where 10^1.67 ~47.6

For the 9 prime repunits the average gap is log(2700343/2)/8 ~0.64 where 10^0.64 ~5,84
But between A(9) and a potential A(10) there is —at the very least— a logarithmic gap of 1.13

Between mersenne primes (a generalized repunit base 2) the largest logarithmic gap is between A₁₂ and A₁₃ and is 0.61 . The theoretical mean ratio is 2^(1/e^∂) or about 1.48 . Does anyone know what is the theoretical ratio between prime repunits?

Seems there is scarcity of repunits compared to other generalized repunits.

In binary, there is no 8-bit and 9-bit prime p such that the repunit 2^p-1 is prime, but for all 2<=k<=27 except 8, 9 and 19, there is k-bit prime p such that 2^p-1 is prime. (the largest 7-bit prime is 127, and the smallest 10-bit prime is 521, both of them satisfy that 2^p-1 is prime) I conjectured that for all k>=2 except 8, 9 and 19, there is k-bit prime p such that 2^p-1 is prime.