proof of Lagrange’s four-square theorem
The following proof is essentially Lagrange’s original, from around 1770. First, we need three lemmas.
For any integers ,
This is the Euler four-square identity, q.v., with different notation.
If is a sum of two squares, then so is .
If is an odd prime, then for some integers with .
Let . Consider the sets
We have the following facts:
Similarly, no two elements in are congruent mod .
Furthermore, since elements of are all non-negative, while elements of are all negative.
Therefore, has , or elements.
Therefore, by the pigeonhole principle, two elements in must be congruent mod . In addition, by the first two facts, the two elements must come from different sets. As a result, we have the following equation:
for some . Clearly is positive. Also, , so . ∎
Basically, Lemma 3 says that for any prime , some multiple of is a sum of four squares, since .
Proof of Theorem.
By Lemma 1 we need only show that an arbitrary prime is a sum of four squares. Since that is trivial for , suppose is odd. By Lemma 3, we know
for some with . If , then we are done. To complete the proof, we will show that if then is a sum of four squares for some with .
If is even, then none, two, or all four of are even; in any of those cases, we may break up into two groups, each group containing elements of the same parity. Then Lemma 2 allows us to take .
Now assume is odd but . Write
where are all in the interval . We have
So for some integer non-negative . Since , . In addition, if , then , so that , which implies , or that . But is prime, forcing , and contradicting . So . Look at the product and examine Lemma 1. On the left is . One the right, we have a sum of four squares. Evidently three of them
are multiples of . The same is true of the other sum on the right in Lemma 1:
The equation in Lemma 1 can therefore be divided through by . The result is an expression for as a sum of four squares. Since , the proof is complete. ∎
Remark: Lemma 3 can be improved: it is enough for to be an odd number, not necessarily prime. But that stronger statement requires a longer proof.
|Title||proof of Lagrange’s four-square theorem|
|Date of creation||2013-03-22 13:21:07|
|Last modified on||2013-03-22 13:21:07|
|Last modified by||CWoo (3771)|