Koblitz states in his book on p-adic numbers on page 84:
Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all primes $p$ for which the binomial series for $(1 + \alpha)^{1/2}$ converges in $|\cdot|_p$. There is no $\alpha$ other than $8$, $16/9$, $3$, $5/4$ for which $(1 + \alpha)^{1/2}$ converges to the same value in $|\cdot|_p$ for all $p \in S$.
Why is this an example of a very general theory of E. Bombieri? What does the theory (or theorem) say and what are its possible connections with p-adic numbers? Is it the Bombieri–Vinogradov theorem from analytic number theory?