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Old 2018-12-19, 10:59   #122
enzocreti
 
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Default ...continue...

moreover pg(36) is prime
pg(36*1935=69660) is prime...36+1=37 is prime...69660+1=69661 is prime
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Old 2018-12-19, 13:01   #123
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Quote:
Originally Posted by Batalov View Post
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
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Old 2018-12-19, 14:24   #124
enzocreti
 
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Default 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
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Old 2018-12-19, 15:19   #125
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Quote:
Originally Posted by enzocreti View Post
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
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Old 2018-12-20, 05:19   #126
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Originally Posted by axn View Post
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.
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Old 2018-12-20, 05:46   #127
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Originally Posted by CRGreathouse View Post
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.
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Old 2018-12-20, 09:41   #128
enzocreti
 
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Default 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
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Old 2018-12-20, 11:06   #129
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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
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Old 2018-12-20, 11:21   #130
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Default pg(19k)

pg(19k) is prime only when 19k is congruent to 19 mod 10
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Old 2018-12-20, 16:18   #131
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Quote:
Originally Posted by enzocreti View Post
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?!?
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Old 2018-12-20, 16:29   #132
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Quote:
Originally Posted by enzocreti View Post
pg(19k) is prime only when 19k is congruent to 19 mod 10
aka when k is 1 mod 10 ...
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