## Tuesday, August 30, 2005

### Useful Equations

I thought it would be useful to review some basic equations. These are presented as a series of lemmas.

Lemma 1: (a + b)2 = a2 + 2ab + b2

(a + b)2 = (a + b)(a + b) = a(a + b) + b(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2.

QED

Lemma 2: (a + b)(a - b) = a2 - b2

(a + b)(a - b) = a(a - b) + b(a - b) = a2 - ab + ab - b2 = a2 - b2

Corollary 2.1: a4 - b4 = (a - b)(a + b)(a2 + b2)

By Lemma 2, a4 - b4 = (a2 - b2)(a2 + b2)

Applying Lemma 2 again, gives us:
a4 - b4 = (a - b)(a + b)(a2 + b2)

Lemma 3: a3 + b3 = (a + b)(a2 - ab + b2)

(a + b)(a2 - ab + b2 ) = a(a2 - ab + b2) + b(a2 - ab + b2) =
a3 - a2b + ab2 + a2b -ab2 + b3 = a3 + b3

Corollary 3.1: a3 - b3 = (a - b)(a2 + ab + b2)

Let b' = -b so that a3 - b3 = a3 + (b')3

By Lemma 3: a3 + (b')3 = (a + b')(a2 - ab' + (b')2) = (a - b)(a2 + ab + b2)

QED

Lemma 4: a5 + b5 = (a + b)(a4 - a3b +a2b2 - ab3 + b4)

(a + b)(a4 - a3b +a2b2 - ab3 + b4) =
=a(a4 - a3b +a2b2 - ab3 + b4) + b(a4 - a3b +a2b2 - ab3 + b4) =
=a5 - a4b + a3b2 - a2b3 + ab4 + a4b - a3b2 + a2b3 - ab4 + b5 =
=a5 + b5

QED

Lemma 4: n ≥ 5 → an + bn = (a + b)(a(n-1) - a(n-2)b + ... - ab(n-2) + b(n-1))

(a + b)(a(n-1) - a(n-2)b + ... -ab(n-2) + b(n-1)) =

a(a(n-1) - a(n-2)b + a(n-3)b2... -ab(n-2) + b(n-1)) +
b(b(n-1)+a(n-1) - a(n-2)b + ...+a2b(n-3) -ab(n-2)) =

an - a(n-1)b + a(n-2)b2 + ... -a2b(n-2) + ab(n-1) +
bn + a(n-1)b - a(n-2)b2 + ... + a2b(n-2) - ab(n-1) =
an + bn

QED

Lemma 5: a - b divides an - bn

Proof:

(1) Let n = be the highest number where this is true.

(2) We know that n is at least 2 since a2 - b2 = (a - b)(a + b)

(3) To complete this proof, we need to show that a - b divides an+1 - bn+1

(4) We know that (a-b)(an) = an+1 - anb and we know that (a-b)(bn) = abn - bn+1

(5) So that an+1 - bn+1 - (a - b)(an) - (a-b)(bn) = anb - abn = ab(an-1 - bn-1)

(6) So that:

an+1 - bb+1 = (a-b)(an + bn) - ab(an-1 - bn-1)

(7) Since n ≥ 2, we know that a-b divides an-1 - bn-1.

QED

Scouse Rob said...

In Corollary 2.1:
There is a missing bracket on the end of (a2 - b2)(a2 + b2

In Lemma 5:
The text get very small...:-)

Rob

Larry Freeman said...

Hi Rob,

Fixed both issues. Thanks very much for posting! :-)

-Larry