Finding all the integer solutions

We have $$(x-1)^3< x^3-2x^2-1 <x^3$$ for all $x\notin \{0,1,2,3\}$

So the only solution for $x$ are in $\{0,1,2,3\}$


Consider $$(x-1)^3\leq y^3<x^3.$$ The right inequality is true for all integer $x$.

The left inequality is true for $x(x-3)\geq0.$

Tags:

Number Theory

Polynomials

Diophantine Equations

Elementary Number Theory

Integers

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