Monday, October 10, 2005

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

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