Basic Lemmas Needed for FLT: n = 5

Here are some basic lemmas that are used by the proof for Fermat's Last Theorem: n=5.

Lemma 1: (p+q)⁵ + (p-q)⁵ = 2p(p⁴ + 10p²q² + 5q⁴)

(1) Using the Binomial Theorem:
(p + q)⁵ = p⁵ + 5p⁴q + 10p³q² + 10p²q³ + 5pq⁴ + q⁵
(p - q)⁵ = p⁵ - 5p⁴q + 10p³q² - 10p²q³ + 5pq⁴ - q⁵

(2) Adding these two values together gives us:
2p⁵ + 20p³q² + 10pq⁴ = 2p(p⁴ + 10p²q² + 5q⁴)

QED

Lemma 2:

If:

t = q⁴ + 50q²r² + 125r⁴
u = q² + 25r²
v = 10r²

Then:

t = u² - 5v²

(1) u² = (q² + 25r²)² = q⁴ + 50q² r² + 625r⁴

(2) -5v² = -5(10r²)² = -500r⁴

(3) (q² + 25r²)² + -5(10r²)² = q⁴ + 50q² r² + 625r⁴ + -500r⁴ = q⁴ + 50q² r² + 125r⁴

QED