<s>[INST] Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed) 
Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer.

I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
 [/INST] That $\,47\,$ is a factor can be verified by very easy mental arithmetic:
$\begin{eqnarray} 
{\rm mod}\ 47\!:\quad && 1+ \color{#c00}{7^8} +\, 8^9 +\ \color{#0a0}{9^7}\\
\equiv && 1+  \color{#c00}{2^4} + 2^{27} + \color{#0a0}{7\cdot 7^6\cdot 8^7}\quad {\rm by}\ \ \color{#0a0}{9\equiv 7\cdot 8},\ \ \color{#c00}{7^2\equiv 2}\\
\equiv&& 1+ 2^4 + 2^{27} + 7\cdot \color{#c00}{2^3}\cdot 2^{21}\\
\equiv&& 1+ 2^4 + 2^{4} +\ 7\cdot 2\,\equiv\, 0\quad {\rm by}\ \ \color{#c00}2^{23}\equiv (\color{#c00}{7^2})^{23}\equiv 7^{46}\equiv 1,\ \ \rm by\ little \ Fermat\\
\end{eqnarray}$
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