Fermat's Last Theorem: n = 3: Division of a2 + 3b2 by 2

In today's blog, I will go over the case of where a² + 3b² is an even number. This result is part of the proof of Fermat's Last Theorem for n=3. If you would like to see the full proof, start here.

Lemma: if 2 divides a polynomial of form a² + 3b², then 4 also divides the polynomial and this division by 4 results in another polynomial of the same form.

In other words, we will prove two propositions:

• if 2 divides a² + 3b², then 4 also divides it.
• if 4 divides a² + 3b², then there exists c,d such that a² + 3b² = 4*(c² + 3d²)

(1) We know that a,b have the same parity (they are both odd or they are both even). Otherwise, a² + 3b² would not be divisible by 2

(2) If they are both even, then it is easy to show that 4 divides a² + 3b² and that the result is of the same form:

(a) There exists c,d such that a = 2c and b = 2d
(b) a² + 3b² = (2c)² + 3(2d)² = 4(c² + 3d²)

(3) So, we can assume that they are both odd.

(4) Now, there exists m,n such that a = 4m ± 1, b = 4n ± 1 [See here for proof]

(5) From this, we know that 4 divides either a + b or a - b

(6) I will now show that in both cases, the lemma is proven.

Case I: 4 divides a + b

(a) First, 4(a² + 3b²) = (1² + 3*1²)(a² + 3b²) = (a - 3b)² + 3(a + b)² [See here for proof]

(b) Since a - 3b = (a + b) - 4b, we know that 4 divides a - 3b

(c) So, this gives us that 4² divides (a - 3b)² + 3(a + b)² which also means that:
4 divides a² + 3b² [Since 4² divides (a - 3b)² + 3(a+b)² implies 4² divides 4(a² + 3b²)]

(d) Since 4 divides a - 3b and a + b, we know that there exists u,v such that: u = (a - 3b)/4 and v = (a + b)/4

(e) 4(u² + 3v²) = 4{[(1/4)(a - 3b)]² + 3[(1/4)(a + b)]²} =
= (1/4)(a² + 9b² + 3a² + 3b²) =
= (1/4)(4a² + 12b²) = a² + 3b²

Case II: 4 divides a - b

(a) The argument here is the same as for case I except that we set 4(a² + 3b²) = (1² + 3*(-1)²)(a² + 3b²) = (a + 3b)² + 3(a - b)² [See here for proof.]

(b) Then after proving that 4² divides this result, we let u = (1/4)(a + 3b) and let v = (1/4)(a - b).

(c) Then 4(u² + 3v²) = a² + 3b²

QED