Erdős–Moser Equation Proof (Part $2$ - The Facts)

In Part $1$ of this series, I said (in detail) that we are going to attempt to prove that

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

has more than $1$ solution using the equation

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

as

$1^k = 1$

Now, we need to lay down the foundations.

Using the first constraint,

$k = 1, 2$

as $x$ needs to be whole.

The second and fifth constraints would contradict the final constraint, hence no need for them.

The first half of the third constraint is useful - using OEIS, the LCM of the first $200$ natural numbers is

$337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000$

The second half of the third constraint is to be used in the proof.

The fourth constraint is obsolete due to the final constraint.

As for the final constraint, we know that

$m > 2.73129 \times 10^{1,667,658,416}$

But what's the value of $k$?

Going into an AMS paper referenced by the Wikipedia page of the Erdős–Moser equation (here is the paper: Erdős–Moser Equation Final Constraint Computations and Proofs Paper),

$k \geq 2$

therefore

$k = 2$

We've got our values of $m, k$.

We're ready to do some algebraic proof!