Problem 718

Unreachable Numbers

Consider the equation $17^pa+19^pb+23^pc = n$ where $a$, $b$, $c$ and $p$ are positive integers, i.e. $a,b,c,p>0$.

For a given $p$ there are some values of $n>0$ for which the equation cannot be solved. We call these unreachable values.

Define $G(p)$ to be the sum of all unreachable values of $n$ for the given value of $p$. For example $G(1) = 8253$ and $G(2)= 60258000$.

Find $G(6)$. Give your answer modulo $1\ 000\ 000\ 007$.

考虑等式$17^pa+19^pb+23^pc = n$，其中$a$、$b$、$c$和$p$均为正整数。

对于给定的$p$，有些取值$n>0$会使得方程无解。我们称这些取值为不可达数。

记$G(p)$为对于给定$p$的所有不可达数$n$之和。例如，$G(1) = 8253$，$G(2)= 60258000$。

求$G(6)$，并将你的答案对$1\ 000\ 000\ 007$取余。