Order of periodic element of general linear group over integers is bounded

Statement
Suppose $$n$$ is a natural number. Then, the order of any element of the fact about::general linear group $$GL(n,\mathbb{Z})$$ having finite order is bounded by a function of $$n$$.

Explicitly, if there exists an element of order $$m$$, then we can write $$n = a_1\varphi(m_1) + a_2\varphi(m_2) + \dots + a_r\varphi(m_r)$$ where $$\varphi$$ denotes the Euler totient function, all $$a_i \ge 1$$, and the lcm of $$m_1,m_2,\dots,m_r$$ equals $$n$$.

This gives a bounding function of the form: the maximum possible order of a $$n \times n$$ matrix is at most $$(1 + n)^{2n}$$. In practice, the maximum possible order is much lower.

Proof of the Euler totient function partition condition
Given: An element $$g \in GL(n,\mathbb{Z})$$ of order $$m$$.

To prove: There exists a partition $$n = a_1\varphi(m_1) + a_2\varphi(m_2) + \dots + a_r\varphi(m_r)$$ where the lcm of $$m_1,m_2,\dots,m_r$$ is $$m$$ and $$\varphi$$ is the Euler totient function.

Proof:

Proof of bounding
Given: $$n = a_1\varphi(m_1) + a_2\varphi(m_2) + \dots + a_r\varphi(m_r)$$, where $$a_1,a_2,\dots,a_r$$ are all positive integers, and the lcm of $$m_1,m_2,\dots,m_r$$ is $$m$$.

To prove: $$m \le n^{2n}$$.