A125607: proof of an irrational constant?

[Nick Hobson]

In my recent submission of A125607 I conjectured that the decimal constant  
formed by concatenating the terms is irrational.  It seems fairly clear  
that the constant will be irrational, though I don't have a proof.  For it  
to be rational would require that the digits recur from some point on,  
which would imply that eventually the order in which successive primes are  
congruent to certain values mod 8, 10, 12, 24, and 28 repeats  
indefinitely, which seems absurd!  In fact, would this not contradict  
Dirichlet's asymptotic results regarding the relative frequencies of  
primes of the form an + b, where gcd(a, b) = 1?  It seems it should, but I  
can't quite pin it down.  If anybody has a concise proof please submit it  
as a comment!

Nick