PRP 14388 (show) (130^6889-1)/(130^83-1)<14388> = 3151053670...01<14388>

P 4490 (show) (306^1849-1)/(306^43-1)<4490> = 1632661778...13<4490>

Hello Peter

Update: primality test finished for $b \le 500$ and $100 < p < 200$.

$10\ 500$ candidates for which after presieving $4\ 168$ remained to test with LLR.

The poor result are the three PRP's mentioned above yesterday.

The other test is still running at $b=521$ in the range $\frac {b}{2} - \sqrt{b} \le p \le \frac {b}{2} + \sqrt{b}$.

In the meantime, I doubt whether there is a higher prime density in this area.

No further PRP's.

@MartinHopf Well done ! I was not successful either with PRP of considerable magnitude.

A bit surprising that there are no more primes considering that no prime factor less than $p^2$ can occur.

Yes, also the pre sieving methods are not as efficient as we could expect.

Actually I'm reading a paper from Yves Gallot:

`CYCLOTOMIC POLYNOMIALS AND PRIME NUMBERS - Yves Gallot´

Any result useful for our search ?

At least some interesting prime numbers should occur in cyclotmic polynomials. I have to understand their densities.

Another milestone should be the range $2\le b\le 1\ 000$ , $2\le p\le 100$ and another one $2\le b\le 10^4$ , $2\le p\le 50$. Do you think we can do them in a reasonable amount of time ?

And finally, there are several primes $p$ left for which there is no known base $b$ leading to a prime.

@Peter I downloaded PFGW today. Very useful, it can handle cyclotmic polynomials.

Yes, but it is of course slower.

Do you have an idea how we can get some more helpers and how to organize the search to get optimal results ?

Phi handles both the Mersenne $b^p-1$ and the Fermat $\large b^{2^n}+1$ ones.

Yes, flexible and useful ! We can even create the ec-numbers because there is a len-command.

I think we should first make a reseach how the primes of yclotmic polynomials

A mini-project : Which base $b$ gives a prime for $p=61$ ? According to my search, no $b\le 4\ 000 $ does the job.

* are researchd

In fact, this could save much time !

Although we invested much time already ...

@Peter $p=61$ is a 'bad' prime but there should be a $b$ that is satisfying.

There are more "bad" primes.

Only "bad" for our special purposes.

of course

Have you found any indication that the $p^2$-th polynomials have been deeply verified for prime values ?

No, let's gather some more information on cyclotmic polynomials.

They are all irreducible and I today learnt that they all apply to Bunyakovsky's conjecture, so should produce infinite many primes.

Another question : If a number must have prime factors of the form $2kn+1$ for a fixed $n$, by which factor is the chance for a prime increased ?