14. Given that $c$ is a positive integer not exceeding 2017, and there exist positive integers $a$ and $b$ such that
$$
2^{a} \times 5^{b}=\left(a^{3}+a^{2}+a+1\right) c \text {. }
$$

Find the maximum value of $c$.
14. According to the problem, we have
$$
\begin{array}{l}
2^{a} \times 5^{b}=\left(a^{3}+a^{2}+a+1\right) c \\
=(a+1)\left(a^{2}+1\right) c .
\end{array}
$$

Thus, the prime factors of \(a+1\) and \(a^{2}+1\) can only be 2 or 5.

Since \(a^{2}+1=(a-1)(a+1)+2\), the greatest common divisor of \(a+1\) and \(a^{2}+1\) is equal to the greatest common divisor of \(a+1\) and 2 (which is 1 or 2).
(1) If \(a\) is even, then \(5 \mid (a+1)\) and \(5 \mid (a^{2}+1)\), which is a contradiction.
Therefore, \(a\) is not even.
(2) If \(a\) is odd, then \(a \neq 1\). Otherwise, \(5^{b}=2 c\), which is a contradiction.
Therefore, \(a \geqslant 3\).
Since \(a^{2}+1\) modulo 4 is 2, we can set
$$
a^{2}+1=2 \times 5^{k}, \quad a+1=2^{m},
$$

where \(k \leqslant b\) is a positive integer, and \(m \geqslant 2\) is a positive integer.
(i) When \(m=2\), we get \(a=3\), \(c=5^{b-1}\).
$$
\text{Let } 5^{b-1} \leqslant 2017 \Rightarrow b \leqslant 5.
$$

At this time, \(c \leqslant 625\).
(ii) When \(m \geqslant 3\), we get
\(\left(2^{m}-1\right)^{2}+1=2 \times 5^{k}\)
\(\Rightarrow 2^{m}\left(2^{m-1}-1\right)=5^{k}-1\)
\(\Rightarrow 81\left(5^{k}-1\right) \Rightarrow k\) is even.
Let \(k=2 l\) (where \(l\) is a positive integer).
Then \(2^{m}\left(2^{m-1}-1\right)=\left(5^{l}-1\right)\left(5^{l}+1\right)\).
Since \(5^{l}+1\) modulo 4 is 2, we can set \(5^{l}-1=2^{m-1} d\) (where \(d\) is a positive odd number).
Thus, \(2^{m}\left(2^{m-1}-1\right)=2^{m-1} d\left(2^{m-1} d+2\right)\), i.e., \(2^{m-1}-1=d\left(2^{m-2} d+1\right)\).
From the above equation, we know \(d2^{m-1}-1\),
$$

which is a contradiction.
Therefore, \(d=1\), i.e., \(2^{m-1}-1=2^{m-2}+1\).
Thus, \(m=3\). Solving, we get \(a=7\), \(c=8 \times 5^{b-2}\).
Let \(8 \times 5^{b-2} \leqslant 2017 \Rightarrow b \leqslant 5\).
At this time, \(c \leqslant 1000\).
In summary, when \(a=7\), \(b=5\), \(c\) reaches its maximum value
1000.