I was reading an article on the well ordering principle and there was a problem that asked to use the well ordering principle to solve Lehman's Lemma:
That there are no positive integer solutions to the following equation:
$$8a^4+4b^4+2c^4=d^4$$
What I want to do is set up a counterexample set $C$ and assume that this set is non-empty. By the well ordering principle there's a least element $a\in C$. How should I use this to reach a contradiction?
Best Answer
Assume that there are positive solutions. Then there is a solution $(a,b,c,d)$ with the smallest possible value $abcd > 0$.
Considering the equation mod $2$, you see that $d$ is even, so $d = 2\delta$, and division by $2$ gives $4a^4 + 2b^4 + c^4 = 8\delta^4$. Now mod $2$ we find that $c$ is even, so $c = 2\gamma$. Going on like this, also $b = 2\beta$ and $a = 2\alpha$ are even, and we get the solution $(a/2,b/2,c/2,d/2)$ which contradicts the minimality condition on $(a,b,c,d)$.