541456 and 51456. I checked 20 numbers 2000 times and found 200 patterns!!

[enzocreti]

moreover pg(36) is prime
pg(36*1935=69660) is prime...36+1=37 is prime...69660+1=69661 is prime

[Dr Sardonicus]

Originally Posted by Batalov

You appear to be unable to distinguish patterns from noise, so you would not be able to see them. You have to learn first.

Here is one pattern for you to start:
31 is prime
331 is prime
3331 is prime
33331 is prime
333331 is prime
3333331 is prime
33333331 is prime
Is 333...331 always prime??
(This is example 6 from R.K.Guy's famous paper.)

These numbers are of the form (10ⁿ - 7)/3. Now,
10^2 = 85 + 15 == -2 (mod 17), so

10^8 = (10^2)^4 == -1 (mod 17), so that

10^9 == -10 (mod 17), giving

10^9 - 7 == 0 (mod 17).

[enzocreti]

pg(56238) and pg(75894) are prp


56238 and 75894 are multiples of 546.


when pg(13k) is prime, then 13k is also a multiple of 42

[science_man_88]

Originally Posted by enzocreti

pg(56238) and pg(75894) are prp


56238 and 75894 are multiples of 546.


when pg(13k) is prime, then 13k is also a multiple of 42

https://www.purplemath.com/modules/inductn.htm

[CRGreathouse]

Originally Posted by axn

Statements of the form "x & y are congruent to k mod p" is utterly meaningless.

Pick any two number x & y.

Factorize x-y. Let p be a prime that divides x-y

x-y == 0 (mod p)
or x == y (mod p)

546-24 = 2*3^2*29

So of course they are both in the same congruent class (mod 29) (and mod 9 and mod 2)

Oh dear. I think you lost him.

[axn]

Originally Posted by CRGreathouse

Oh dear. I think you lost him.

I think you're right. I know when I'm beaten. I accept defeat and graciously bow out.

[enzocreti]

pg(51456), pg(56238), pg(69660) and pg(75894) are three consecutive pg primes.
51456, 56238, 69660 and 75894 are multiples of 6.


Moreover


pg(3336), pg(51456), pg(56238), pg(69660) and pg(75894) are primes and


3336, 51456, 56238, 69660, 75894 are divisible by a prime of the form 6s+1

[enzocreti]

pg(19) and pg(285019) are probable primes, the only ones with 19 and 285019 multiples of 19.


19 and 285019 end with digits 19

[enzocreti]

pg(19k) is prime only when 19k is congruent to 19 mod 10

[Dr Sardonicus]

Originally Posted by enzocreti

pg(19k) is prime only when 19k is congruent to 19 mod 10

Hmm. The 2 values of 19*k have k = 1 and k = 15001. So, all (two) PRP values of pg(19*k) (found so far) have 19*k congruent to 19 (mod 15000), or k == 1 (mod 15000).

But wait -- there's more!!! That second value of k is 15001 = 7*2143.

Now, everyone knows that 22/7 is a good approximation to the number pi -- it's a convergent to the simple continued fraction for pi.

Less well known is that, as discovered by Ramanujan, (2143/22)^(1/4) is quite a good approximation to pi, good to 8 decimal places!

And there are both 7 and 2143 in that second k-value! What does it all mean?!?

[science_man_88]

Originally Posted by enzocreti

pg(19k) is prime only when 19k is congruent to 19 mod 10

aka when k is 1 mod 10 ...