## Extension of Dirchlet Theorem

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• RE: Adam Karasek post for pythagorean triples with a and c primes and b a product of at almost four primes, I seek primes p with 2p-1 and 4p-1 simlutaneously
Message 1 of 2 , Nov 1, 2003
RE: Adam Karasek post for pythagorean triples with a and c primes and
b a product of at almost four primes, I seek primes p with 2p-1 and
4p-1 simlutaneously prime (to reformulate, we could write p=2n+1, and
then seek 2n+1,4n+1,8n+1 simultaneously prime). I am having poor
luck finding such p values, contrary to the belief in the extension
of Dirichlet's theorem. For instance, I generate a random 12 digit
prime, search the through the next 5000 primes, and find about 200-
300 that have 2p-1 prime, but only 0 of those having 4p-1 prime.

Am I not being patient enough or...?

• I dont think any exist besides a few trivial cases. First, p must be congruent to either 1 or 2 mod 3 because if it was congruent to 0 it would be a multiple
Message 2 of 2 , Nov 1, 2003
I dont think any exist besides a few trivial cases.

First, p must be congruent to either 1 or 2 mod 3 because if it was
congruent to 0 it would be a multiple of 3.

so we have two cases:

case 1:
p = 1 mod 3
2p-1 = 1 mod 3
4p-1 = 3 mod 3 = 0, so 4p-1 is a multiple of 3 (not prime)

case 2:
p = 2 mod 3
2p-1 = 3 mod 3 = 0 so 2p -1 is a multiple of 3 (not prime)
4p-1 = 5 mod 3 = 2

thus, this never works unless we take 3 to be that multiple.

so this implies in case 1, p = 1 (not a solution)
or in case 2, 2p - 1 = 3, p = 2
so p = 2, 2p-1 = 3, 4p-1 = 7 so (2,3,7) is a solution

lastly, p could be = 0 mod 3, so p = 3, 2p-1 = 5, 4p-1 = 11
so (3,5,11) is also a solution.

Thus there are only two solutions and your search is futile :)

Lawrence

On Sat, 1 Nov 2003, Adam wrote:

> RE: Adam Karasek post for pythagorean triples with a and c primes and
> b a product of at almost four primes, I seek primes p with 2p-1 and
> 4p-1 simlutaneously prime (to reformulate, we could write p=2n+1, and
> then seek 2n+1,4n+1,8n+1 simultaneously prime).  I am having poor
> luck finding such p values, contrary to the belief in the extension
> of Dirichlet's theorem.  For instance, I generate a random 12 digit
> prime, search the through the next 5000 primes, and find about 200-
> 300 that have 2p-1 prime, but only 0 of those having 4p-1 prime.
>
> Am I not being patient enough or...?
>
>
>