$\exp(\pi\sqrt{163})$ is an integer. Proof: it has a mathematically interesting definition, and the $12$ first digits after dot are zeros.
No integral power of $2$ has $7$ as first digit. Proof: compute by hands successive powers $2n$ 2^n$for an hour. You can't find one beginning with$7$. Well, if you ask a computer, it is a different tale. 1 [made Community Wiki]$\exp(\pi\sqrt{163})$is an integer. Proof: it has a mathematically interesting definition, and the$12$first digits after dot are zeros. No integral power of$2$has$7$as first digit. Proof: compute by hands successive powers$2n$for an hour. You can't find one beginning with$7\$. Well, if you ask a computer, it is a different tale.