How to prove that $4x^3+6x^2+4x+1$ is not a fourth power of an integer, for any $x\in\mathbb N$?

Question

How do I prove that for all positive integers $x$ it's true that, $4x^3+6x^2+4x+1$ is not a fourth power of an integer?

I've tried doing modulo 3 and 5 checks, and it didn't really go far from there

Answer 1

If $4x^3+6x^2+4x+1=w^4$, then: $$ x^4 + w^4 = (x+1)^4 $$ but such an identity contradicts the case $n=4$ of the Fermat's last theorem (there is an elementary proof of that case in terms of Gaussian integers, so the full power of the general FLT is not needed).

Answer 2

Note we can write: y⁴ = 4x³ + 6x² + 4x + 1 as: y⁴ = (x + 1)⁴ - x⁴ and as mentioned is a form of Fermat's Last Theorem. → x⁴ + y⁴ = (x + 1)⁴ This instance of FLT was put to rest long before Wiles finally put the problem to rest. Check out this simple proof: http://fermatslasttheorem.blogspot.com/2005/05/fermats-last-theorem-n-4.html?m=1