TITLE: how to show that the sequence of primes modulo 4 is not eventually periodic?
QUESTION [6 upvotes]: the question as posed is easily seen to be equivalent to asking how to show that a certain number is irrational.
the number referred to as $\Phi$ is defined briefly beneath the question. it is a characteristic number encoding an infinite sequence of binary bits. in this case the information concerns the parity of the odd primes modulo $4$.
question is there a quick demonstration of the irrationality of $\Phi$?

i have a few other curiosities about the distribution of the binary digits of this number (or of the coefficients of the corresponding 2-adic integer), but the simple number theoretic question is all i wish to ask here.
in fact since the deep properties of the primes seem to be encrypted beyond the polynomial domain, one might judge it very likely that $\Phi$ is transcendental, but the assertion of its irrationality is a much more modest claim.  
let $P \subset \mathbb{N}$ be the odd primes, numbered in ascending order so $p_0=3$,  $p_1=5$ and so on. define the parity function on primes $\phi \in 2^{\mathbb{N}}$ by
$$
\phi(n) = \phi_n = \frac12 (1 + i^{p_n-1})
$$
this gives rise to the sequence $\Phi_m$ where:
$$
\Phi_m = \sum_{k=0}^m \frac{\phi_k}{2^k}
$$
with 
$$
\Phi = \lim_{m \to \infty} \Phi_m = 1.010110011... = 1.34960...
$$

REPLY [13 votes]: It is known by a result of Daniel Shiu that there are arbitrarily long runs of consecutive primes congruent to $1$ modulo $4$. (It also would work for $3$ modulo $4$ or in fact any admissible congruence class.) 
Thus, the binary expansion of your number will contain arbitrarily long stings of the same digit and thus cannot be eventually periodic, whence the number is irrational.   
There is some discussion of this and other methods to see this 
in the MO question The prime numbers modulo $k$, are not periodic
Moreover, for what could be called a decimal version of your problem see Number made from ending digits of primes 
In particular, note that the answer to such a question will be the same for any base and minor modifications, as the eventual nonperiodicity is known for any modulus as mentioned above.