I've heard the proof that this number is irrational is accessible to even a novice to number theory:
$\alpha = 0.2 \ 3 \ 5 \ 7 \ 11 \ 13 \ 17 \ldots$
The proof may utilize that a number is irrational iff its decimal expansion either terminates or is periodic, but then I have to show that the set of primes doesn't eventually look something like this:
$\mathbb P = \{\ldots 171, 17171, 171717171, \ldots \}$
I was also told that there is a proof not dependent on that theorem. edit: by "that theorem" the theorem that characterizes irrationals as non-repeating decimals.
Does anyone know simple proofs of this? Thanks.
Hint: If $ \gcd (a, d) = 1$, then there exists infinitely many primes in the arithmetic progression $ a + n d$.
Note: This is a high power theorem whose proof is (arguably) inaccessible to a novice in number theory, but the statement is easy to understand and to accept as fact.
Hint: A number is rational if and only if its decimal representation finally repeats/terminates in 0.
Note: This is a simple fact. I'm not sure if this is the theorem you are referencing in the statement, but it doesn't require "primes eventually look something like this".
Hint: Prove that for any $k$, there must exist a sequence of $k$ 0's in the number.
Hence, this number is irrational.