The alternating sum of primes defines an injection

Question

Define $\displaystyle\alpha(n)=\sum^n_{k=1}(-1)^{n-k}p_k$, where $p_k$ is the $k$:th prime. Show that $\alpha$ is an injection $\mathbb Z_+\to\mathbb Z_+$.

It's easy to see while considering sums as $(3-2)+(7-5)+(13-11)+\cdots$, but I would like a short and nice proof.

Answer 1

1. First, show that $\alpha(2k+2) > \alpha(2k)$ and $\alpha(2k+3) > \alpha(2k+1)$, so that $\alpha$ is injective when restricted to either even or odd $n$.

2. Second, show that $\alpha(2k) \neq \alpha(2l+1)$ for any $k,l$ using a parity argument.