Hi,

I have been investigating some old buddies in the realm of prime numbers. If

we take for example

the first 999 digits of Pi and append that to the sequence of numbers below

we will get a greater

than 1000 digit pseudoprime. Pari code is at the end to generate these and

the pprimes thenselves.

61,1158,2488,2623,2625,3454,4575,5100,8019,8821,...

Notice 2623,2635. What, a kind of twin prime? Are there more of these?

Dirichlet proved in 1837 that every arithmetic progression kn + a where

(k,a)=1, n=1,2,3,..

contains an infinite number of primes. The Full proof can be found in

Lectures on Elementary

Number Theory by Hans Rademacher pp121 - 135.

If we set k = 10^999 and a = 314159...4209. then the terms in the above

sequence are infinite.

If we set n=1,k=1 we have the progression 1+a. So for a=1,2,3,..

Dirichlet's theorem proves

the infinity of primes per se.

My question is this. Why can't we set n = 0 in the Dirichlet theorem once

proved? Dirichlet used

zeta(s) = Sum(n=1..infinity,1/n^s) in his proof. This restricts n=0 unless

of course we let s=0 and

this would mean zeta(s) = infinity which dosen't tell us much or too much

depending on how you

look at it. Nevertheless, the finished product, kn + a will accept n=0

without restriction so the

progression of integers 0 + a = 1,2,3... contain an infinite number of

primes. I don't see a catch 22

here since "that which is true is true and cannot be changed." But...

Pari Code

f(n) =

e=floor(Pi*10^998);for(x=0,n,z=x*10^999+e;if(ispseudoprime(z),print1(x",")))

Have fun in the facinating world of numbers,

Cino

"Behavior is not for the pursuit of survival but because of it."

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