Erdős–Moser Equation Proof (Part $1$ - Setting the Boundaries and Definition)

In number theory, the Erdős–Moser equation is defined as

$1^k + 2^k + ... + m^k = (m + 1)^k$

Of course, a well-known trivial solution is

$1^1 + 2^1 = 3^1$

However, Paul Erdős conjectured that no further solutions exist to the Erdős–Moser equation other than

$1^1 + 2^1 = 3^1$

So, now what?

Well, a solution to find $m$ is

$m^k - (m + 1)^k = - 1 - 2^k$

since

$1^k = 1$

Leo Moser also posed certain constraints on the solutions:

$1$. $\frac{2}{k} = x$, where $x$ is a positive number and that there is no other solution for $m < 10^{1,000,000}$, which Leo Moser himself proved in $1953$.

$2$. $k + 2 < m < 2k$, which was shown in $1966$.

$3$. $lcm (1, 2, ..., 200)$ divides $k$ and that any prime factor of $m + 1$ must be irregular and $> 10,000$, which was shown in $1994$.

$4$. In $1999$, Moser's method was extended to show that $m > 1,485 \times 10^{9,321,155}$ .

$5$. In $2002$, it was shown that $200 < p < 1000$ must divide $k$, where $p$ is all the primes in between.

$6$. In $2009$, it was shown that $\frac{2k}{2m - 3}$ must be a convergent of $In (2)$ - large-scale computation was used to show that $m > 2.7139 \times 10^{1,667,658,416}$.