four-squaretheorem(lagrange)

Four-Square Theorem (Lagrange)

Any positive integer can be written as the sum of four squares.

Proof:

From Lemma 1 (Euler) and Lemma 2, by induction.

Lemma 1

If the integers m and n are each the sum of four squares, then mn is also.

Proof:

m=a1**2+a2**2+a3**2+a4**2, n=b1**2+b2**2+b3**2+b4**2

mn=(a1**2+a2**2+a3**2+a4**2)(b1**2+b2**2+b3**2+b4**2)

=(a1b1+a2b2+a3b3+a4b4)**2+(a1b2-a2b1+a3b4-a4b3)**2+(a1b3-a2b4-a3b1+a4b2)**2+(a1b4+a2b3-a3b2-a4b1)**2

Lemma 2

Any prime p can be written as the sum of four squares.

Proof:

2=1**2+1**2+0**2+0**2