Infinite Descent

Infinite Descent is a method of proof which utilizes the extremal principle to contradict the extremality of an extreme object. This principle is best described using an example:

Problem 1. Prove that $\sqrt 2$ is irrational.

Proof. Assume otherwise that $\sqrt 2$ is rational; that is, $\sqrt 2$ =$\dfrac{p}{q}$, where p and q are positive integers. Squaring both sides gives $2=\dfrac{p^2}{q^2}$, and $2q^2 = p^2$. Now, we assumed that there is at least one solution of (p, q). Among all possible solutions, let (P, Q) be the solution with P + Q minimized. Then $2Q^2 = P^2$. Clearly P is not odd; thus, it is even and equals 2P'. Then $2Q^2 = 4P'^2$ and $Q^2 = 2P'^2$, or by symmetric property $2P'^2 = Q^2$. Thus, (Q, P') is a solution to $2q^2 = p^2$. But $P' + Q = \dfrac{P}{2}+ Q < P + Q$, contradicting the fact that $(P, Q)$ is the solution with $P + Q$minimized. Thus, there is no solution in positive integers to $\sqrt 2$ = $\dfrac{p}{q}$, and hence $\sqrt 2$ is irrational.

Notice how in this proof we looked at the (p, q) that had certain properties (in this case the sum of the numbers) and found another solution which had properties of greater degree (in this case a smaller sum) than the previous one. In essence, infinite descent relies on the fact that one can repeat the process of finding a solution of greater degree infinitely, which is obviously impossible.

Try your hand at the following problems:

Problem 2. Prove that $9a^3 + 3b^3 + c^3 = 0$ has only one solution in integers, (0, 0, 0).

Problem 3. (Fermat's Last Theorem for n = 4): Prove that $a^4 + b^4 = c^4$ has no solutions in positive integers.

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