Proof for the existence of primes not equal to $ap_\alpha +bp_\beta$ etc?

Question

Is there a general proof to show that there exists prime numbers larger than $min(p_\alpha,p_\beta)$that are not equal to $ap_\alpha +bp_\beta$, given $p_\alpha,p_\beta\in\mathbb{P}-\left\{2\right\}$ and the condition that $a,b\in\mathbb{N}_0 $?

Answer 1

Your question should be phrased as whether there are primes greater than $C = \min(p_\alpha, p_\beta)$ which cannot be represented in the form. Any prime smaller than $C$ certainly cannot be represented in the form. For sufficiently large numbers $n$, $n$ can be represented in the given form so the question really boils down looking for a counter-example among numbers that are greater than $C$ but not "too large". A weaker version of the twin prime conjecture says something like you can find arbitrary large pairs of primes $p,p+k$ where $k$ is bounded by some constant I can't remember. If $p >> k$ and $p$ is large then you can then use a result that says there is a prime $q$ in between $p+k$ and $1.5(p+k)$ (actually I believe the constant is sharper than $1.5$ but I can't remember it) and this prime will not be representable as a non-negative linear combination of the primes $p$ and $p+k$.

Answer 2

Without loss of generality we may assume that $p_\alpha\lt p_\beta$.

Suppose first that $p_\beta\gt 2p_\alpha$. Then by Bertrand's Postulate there is a prime $x$ such that $p_\alpha\lt x\lt 2p_\alpha$. This prime cannot be represented in the desired form.

Suppose next that $p_\beta\lt 2p_\alpha$. By Bertrand's Postulate, there is a prime $y$ such that $p_\beta\lt y\lt 2p_\beta$. To represent $y$, we would have to use $1$ copy of $p_\beta$, and at least $2$ copies of $p_\alpha$. But $2p_\alpha+p_\beta\gt 2p_\beta$.