Axel Thue’s proof of Fermat’s theorem on sums of two squares

Topics:
maths

Fermat’s theorem on sums of two squares is a mainstay of undergraduate maths courses:

**Theorem.** If p is a prime number that is congruent to 1 modulo 4, then p is a sum of two integer squares:
`p = a^{2} + b^{2}`

.

Fermat announced the theorem in 1640, but didn’t supply a proof.
Euler gave the first complete proof in 1747.
Further proofs have been given by many others, including Lagrange and Dedekind.
I first saw the theorem proved as an application of Minkowski’s lattice theorem, although this is surely overkill.
There is now even a ‘one-sentence proof’ due to Don Zagier.^{1}
For my money though, the best proof is due to the Norwegian mathematician Axel Thue (1863–
1922).^{2} Thue’s proof rests on the following well-known lemma:

**Lemma.** If p is a prime number that is congruent to 1 modulo 4, then there is a `u`

such that
`u^{2} \equiv -1 \mod p`

.

The lemma follows easily once you know that the group of units of a finite field is cyclic. Alternatively one can give a direct proof using Wilson’s theorem.

*Proof of the theorem.*

Let `k`

be the integer part of `\sqrt{p}`

, so that `k^{2} < p < (k+1)^{2}`

, and let `u`

be a square root of -1 modulo `p`

as above.
We consider the values `a - ub \mod \quad p`

, where `0 \le a,b \le k`

.
There are `(k+1)^{2}`

such pairs `(a,b)`

and only `p`

congruence classes, so there exist distinct pairs `(a_1,b_1) \ne (a_2,b_2)`

such that `a_1 - ub_1 \equiv a_2 - ub_2 \mod \quad p`

.
Then by setting `a = a_1 - a_2`

and `b = b_1 - b_2`

we have `a - ub \equiv 0 \mod \quad p`

with `a, b`

not both zero and `|a|, |b| \le k`

.
Now `a^{2} \equiv (ub)^{2} \equiv -b^{2} \mod \quad p`

, so `p`

divides `a^{2}+b^{2}`

.
But `0 < a^{2} + b^{2} \le 2k^{2} < 2p`

, so `a^{2} + b^{2} = p`

.
**QED**.

What I like about this proof is that is short and elementary, but the result doesn’t appear from nowhere as in Zagier’s one-liner. Unlike the proofs of Dedekind or Minkowski, it could be presented in a very first introduction to number theory.

Thue himself is little-remembered these days; his proof of Fermat’s theorem at least deserves to be more widely known.