2

I've already tried to define a set of positive integers that contains all such numbers $c$, i.e $D= \{c : c^3 = 4a^3 + 2b^3, a, b \in \mathbb{Z}^+ \}$, then assuming is not an empty set for the sake of contradiction, if it is not an empty set of nonnegative integers then because of the WOP it must have a minimum element, I called this element $m$, but I can't find a way to prove that no positive integer raised to the third power can be represented by an expression like the given one with $a,b$ also being positive integers. I need help with this one. Thanks beforehand.

Santiago Rodríguez
  • 35
  • https://math.stackexchange.com/questions/3177320/combinatorics-problems-that-can-be-solved-via-infinite-descent – nonuser Sep 15 '19 at 19:21
  • https://math.stackexchange.com/questions/522822/proof-of-the-infinite-descent-principle?noredirect=1&lq=1 – nonuser Sep 15 '19 at 19:22

1 Answers1

5

Let $(a,b,c)$ be a solution with $c$ the smallest possible. Then $c^3$ is even, hence $c$ itself is even. Put $c=2c_0$; then $2a^3+b^3=4c_0^3$. This time, we see that $b$ must be even. Put $b=2b_0$; then $a^3+4b_0^3=2c_0^3$, thus $a=2a_0$ must be even, and $4a_0^3+2b_0^3=c_0^3$, i.e. $(a_0,b_0,c_0)$ is another solution. But $c_0=c/2<c$. Contradiction.

metamorphy
  • 39,111