14506RE: Primes in the concatenation with the digits of Pi
- Feb 2, 2004Hi,
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.
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
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...
Have fun in the facinating world of numbers,
"Behavior is not for the pursuit of survival but because of it."
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