#### TheModMethod

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Prove there is no integer solutions to $x^2 = y^5 - 4$.

Find all integer solutions to: $x^2 + 3y^2 = 1998x$.

Show the equation $x^2 + y^2-8z^3 = 6$ has no integer solution.

Solve in positive integers the equation $3^x + 4^y = 5^z$ .

Solve in positive integers the equation $8^x + 15^y = 17^z$.

Show that neither $385^{97}$ nor $366^{17}$ can be expressed as the sum of cubes of some consecutive integers.

Show that if there exist integer $x$, $y$, and $z$ such that $3^x + 4^y=5^z$, then both $x$ and $z$ must be even.

Find all ordered integer pairs $(x, y)$ such that $x^3 + y^3=2019$.

Find all the integer pairs $(x, y)$ such that $x^3 = 2^y + 15$.

Determine all positive integer $n$ such that the following equation is solvable in integers: $$x^n + (2+x)^n + (2-x)^n = 0$$