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

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.

  1. D. Zagier, A one-sentence proof that every prime p congruent to 1 modulo 4 is a sum of two squares, Am. Math. Monthly 97 (1990), no. 2, p144. ↩︎

  2. A. Thue, Et par antydninger til en taltheoretisk metode, Kra. Vidensk. Selsk. Forh. 7 (1902), pp57–75. ↩︎

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© Tom Harris 2015–2018.

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