Sierpinski number analogue in another base

Question

I was wondering if Sierpinski numbers would work in other bases, and after a little experimentation: yep!

For instance,

$$157 \cdot 23^n - 2$$

is composite for all $n$ using a tiny covering set. Or,

$$5515 \cdot 11^n + 2.$$

If I'm mistaken, please let me know.

Answer 1

$157\times23^{2k+1}-2\equiv1\times2-2=0\bmod 3$,

$157\times23^{4k+2}-2\equiv(-2)(-1)-2=0\bmod53$,

and $157\times23^{4k}-2\equiv2\times1-2=0\bmod5$,

so $157\times23^n-2$ is composite for all $n$.

$5515\times11^{3k+1}+2\equiv5\times11+2\equiv0\bmod 19$,

$5515\times11^{3k+2}+2\equiv6\times16+2\equiv0\bmod7$,

$5515\times11^{2k}+2\equiv1\times1+2\equiv0\bmod3,$

and $5515\times11^{6k+3}+2\equiv2\times(-1)+2\equiv0\bmod37$,

so $5515\times11^n+2$ is composite for all $n$.