mersenneforum.org 541456 and 51456. I checked 20 numbers 2000 times and found 200 patterns!!
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 2018-12-19, 10:59 #122 enzocreti   Mar 2018 2×5×53 Posts ...continue... moreover pg(36) is prime pg(36*1935=69660) is prime...36+1=37 is prime...69660+1=69661 is prime
2018-12-19, 13:01   #123
Dr Sardonicus

Feb 2017
Nowhere

13·373 Posts

Quote:
 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).

Last fiddled with by Dr Sardonicus on 2018-12-19 at 13:01

 2018-12-19, 14:24 #124 enzocreti   Mar 2018 2·5·53 Posts pg(56238) and pg(75894) 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
2018-12-19, 15:19   #125
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts

Quote:
 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

2018-12-20, 05:19   #126
CRGreathouse

Aug 2006

135338 Posts

Quote:
 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.

2018-12-20, 05:46   #127
axn

Jun 2003

120068 Posts

Quote:
 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.

 2018-12-20, 09:41 #128 enzocreti   Mar 2018 53010 Posts exponents leading to a prime 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
 2018-12-20, 11:06 #129 enzocreti   Mar 2018 2·5·53 Posts 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
 2018-12-20, 11:21 #130 enzocreti   Mar 2018 2×5×53 Posts pg(19k) pg(19k) is prime only when 19k is congruent to 19 mod 10
2018-12-20, 16:18   #131
Dr Sardonicus

Feb 2017
Nowhere

484910 Posts

Quote:
 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?!?

2018-12-20, 16:29   #132
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts

Quote:
 Originally Posted by enzocreti pg(19k) is prime only when 19k is congruent to 19 mod 10
aka when k is 1 mod 10 ...

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