Many (most?) number theory proofs employing the method of infinite descent proceed something like this:
- Assume a given [Diophantine] equation (e.g., $x^3+y^3=z^3$) has solutions in positive integers.
- Manipulate the equation until you find an equation of the same form which is an implication/consequence of the first (e.g., $u^3+v^3=w^3$) .
- Show that $0<u<x$ [or $0<v<y$ or $0<w<z$], thus exposing the desired contradiction (because there is not an infinite progression of positive numbers with decreasing absolute value).
I’m wondering if there are any examples of infinite descent proofs which instead proceed like this:
- Assume that a given Diophantine equation has a solution in some positive integer (e.g.) $z=p_1^{\mu_1} p_2^{\mu_2} \dotsb p_k^{\mu_k}$ for primes $p_1 < p_2 < \dotsm < p_k$ and positive integers $\mu_1, \mu_2, \dotsc, \mu_k$ with $k \ge 1$.
- Manipulate the equation to find an equation of the same form with $w=q_1^{\tau_1} q_2^{\tau_2} \dotsb q_m^{\tau_m}$ for primes $q_1 < q_2 < \dotsm < q_m$ and positive integers $\tau_1, \tau_2, \dotsc, \tau_m$ with $m \ge 1$.
- Show that $m < k$, thus proving that $z$ must be a prime power.
- Deliver the coup de grâce of the proof by showing that $z$ cannot be a prime power, or can only be a particular (determinate) prime, or similar.
I haven’t found any such proofs, but my intuition says they must exist. I’m hoping someone out there can provide some references.
