prove that there is no non zero solution to $a^3+b^3=c^3$

Question

Prove that there do not exist three consecutive positive integers a, b, c such that $a^3+b^3=c^3$. How do we proof this using contradiction?

Answer 1

Write $\;a=n-1\,,\,\,b=n\,,\,\,c=n+1\;$ , so you want to know whether$

$$(n-1)^3+n^3=(n+1)^3\iff2n^3-3n^2+3n-1=n^3+3n^2+3n+1\iff$$

$$n^3-6n^2+6n-2=0$$

and now use the rational root of integer polynomial theorem.

Answer 2

Consider the equation mod $4$.

Answer 3

if the order of the integers is $a \lt b \lt c$ then $a$ and $c$ must be odd and $b$ even. so for some $n$ $$ 8n^3 -24n^2 -2 =0 $$ now apply @Robert's hint