7:01 AM
A quick summary with PARI/GP :
gp > forprime(p=3,30,print1(p," ");f=polcyclo(p^2);for(m=2,500,if(ispseudoprime(subst(f,x,m))==1,print1(m," ")));print)
3 2 3 8 11 20 21 26 30 50 51 56 60 78 98 102 117 129 134 146 159 171 186 189 191 198 200 209 210 212 222 240 249 267 269 278 279 299 300 333 344 363 383 390 398 399 425 429 438 444 450 458
5 22 33 39 43 62 74 134 142 167 212 238 287 313 335 369 414 415 418 432
7 2 5 62 67 104 123 138 151 186 194 233 236 261 269 306 359 416 425 439
11 43 174 236 261 293 300 306 312
13 24 120 205 262 408
The case $p=2$ was omitted, this just leads to the primes of the form $b^2+1$

8:00 AM
Hi Peter, thank you for opening this channel !

@MartinHopf Hi
(605^(31^2)-1)/(605^31-1)
(685^(31^2)-1)/(685^31-1)
(729^(31^2)-1)/(729^31-1)

At the moment we are seeking for prime numbers of the form: $\Large \frac{b^{p^2}-1} {b^p-1} = \Large \frac{(b^p)^p-1} {b^p-1} = \Phi_p (b^ {\ p})$
with $p \in \mathbb{P}$ and $\large \Phi_p ()$ is the $p$th cyclotomic polynomial .

(61^(37^2)-1)/(61^37-1)
(659^(37^2)-1)/(659^37-1)
Or , equivalent , the $p^2$ - th cyclotomic polynomial at $x=b$
(697^(41^2)-1)/(697^41-1)
(306^(43^2)-1)/(306^43-1)
(530^(43^2)-1)/(530^43-1)

8:19 AM
@Peter Is this a faster way to get primes than with a pre-sieving method?

No, just a nicer representation.
I currently search an example with $n=47$
By the way, is your range now finished ?
(1153^(47^2)-1)/(1153^47-1)

$p=193$ at the moment. Maybe finished on Sunday. With LLR I test $100 < p < 200$

Which primes/PRP's (other than posted above) can you add to the summary ?
(370^(53^2)-1)/(370^53-1)

8:39 AM
No more PRP's at the moment.
I reached $b=492$ within the search in the range for $\frac {b}{2} - \sqrt{b} \le p \le \frac {b}{2} + \sqrt{b}$.
@Peter If I understand your code right, should be the same as:
forprime(p=3,30,print1(p," ");for(m=2,500,if(ispseudoprime(polcyclo(p,m^p)),print1(m," ")));print)

This is in fact the same.

The computing speed is also the same.

(75^(59^2)-1)/(75^59-1)
Now $n=61$. I am curious which is the smallest example.

9:02 AM
Known PRP's for $p > 100$ are:
and

$n=61$ turns out to be tough. I am at $b=793$

9:25 AM
passed $b=1\ 000$

3 hours later…
12:09 PM
I am at $b=3\ 700$