Fermat's Last Theorem: n = 3 : last lemma needed for proof

It may surprising that we only need one more lemma to finish Euler's proof for Fermat's Last Theorem: n=3. This is an easy one. For those interested in seeing the entire proof for Fermat's Last Theorem: n = 3, start here.

Lemma: For all odd numbers, there exists a value n such that the number is equal to 4n ± 1 with n greater than 0.

(1) Let S = the set of all odd numbers representable by 4n + 1 or 4n - 1

(2) We know that 1 is an element of this set since 1 = 4(0) + 1

(3) We can then presuppose that there is some value v such that all odd numbers less than or equal to v are describeable by 4n + 1 or 4n - 1. We know that v is greater or equal to 1

(4) Now, we will prove that if v is of the form 4n + 1 or 4n - 1, then v + 2 is also of this form.

Case I: v is of the form 4n + 1

v + 2 = 4n + 1 + 2 = 4n + 3 = 4(n+1) - 1

Case II: v is of the form 4n - 1

v + 2 = 4n - 1 + 2= 4n + 1

(5) By the Principle of Mathematical Induction, we have proven this lemma.

QED