Euler tackled several quartic Diophantine equations whose impossibility can be proven by infinite descent. This video revisits that family, extending it slightly, and approaches it without Pythagorean parametrizations, using factorization, coprimality, modular constraints, and simple identities. The focus is on showing how descent becomes cyclical rather than linear, and on revealing connections between these quartic equations.
00:01 Introduction
01:14 Case x^4 + y^4 = z^2
04:13 Case x^4 - 4 y^4 = z^2
08:38 Case x^4 + 8 y^4 = z^2
11:41 Case x^4 - y^4 = z^2
16:36 A family of identities linking Euler's quartic equations
Leonhard Euler - "The proofs of some arithmetic theorems"
https://scholarlycommons.pacific.edu/euler-works/98/
my blog: https://pablonumbertheory.blogspot.com/
# Pablo Pintabona
