Fermat observed that 26 is the only number between a square and a cube... but this “sandwich problem” is just a doorway to introduce primes representations of the forms a² + b² and a² + 2 b² and to set the stage for his method of infinite descent. Here we'll present purely elementary proofs, but don't miss Euler's elegant infinite descent proof for primes 4n+1 as sum of two squares (see link below).
00:02 Introduction
01:16 Case x² + 4 = y³
05:27 Case x² + 2 = y³
All primes 4n+1 are of the form x² + y² — Euler's Proof by Infinite Descent
https://youtu.be/8yobo69-U2k?si=0ZqTruSulD9UrqJa
Also check a related video:
x³ + 1 = y² — Completing Euler's Infinite Descent
https://youtu.be/9Ulfio1XcIg?si=tNxtyWVnPBnaiN8Q
my blog: https://pablonumbertheory.blogspot.com/
# Pablo Pintabona
