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HandWiki. Proofs of Fermat's Theorem on Sums of Two Squares. Encyclopedia. Available online: https://encyclopedia.pub/entry/33999 (accessed on 20 September 2026).
HandWiki. Proofs of Fermat's Theorem on Sums of Two Squares. Encyclopedia. Available at: https://encyclopedia.pub/entry/33999. Accessed September 20, 2026.
HandWiki. "Proofs of Fermat's Theorem on Sums of Two Squares" Encyclopedia, https://encyclopedia.pub/entry/33999 (accessed September 20, 2026).
HandWiki. (2022, November 11). Proofs of Fermat's Theorem on Sums of Two Squares. In Encyclopedia. https://encyclopedia.pub/entry/33999
HandWiki. "Proofs of Fermat's Theorem on Sums of Two Squares." Encyclopedia. Web. 11 November, 2022.
Proofs of Fermat's Theorem on Sums of Two Squares
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Fermat's theorem on sums of two squares asserts that an odd prime number p can be expressed as with integer x and y if and only if p is congruent to 1 (mod 4). The statement was announced by Girard in 1625, and again by Fermat in 1640, but neither supplied a proof. The "only if" clause is easy: a perfect square is congruent to 0 or 1 modulo 4, hence a sum of two squares is congruent to 0, 1, or 2. An odd prime number is congruent to either 1 or 3 modulo 4, and the second possibility has just been ruled out. The first proof that such a representation exists was given by Leonhard Euler in 1747 and was complicated. Since then, many different proofs have been found. Among them, the proof using Minkowski's theorem about convex sets and Don Zagier's short proof based on involutions have appeared.

involutions perfect square prime number

References

  1. Euler à Goldbach, lettre CXXV http://www.math.dartmouth.edu/~euler/correspondence/letters/OO0852.pdf
  2. De numerus qui sunt aggregata duorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 4 (1752/3), 1758, 3-40) [1]
  3. Demonstratio theorematis FERMATIANI omnem numerum primum formae 4n+1 esse summam duorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 5 (1754/5), 1760, 3-13) [2]
  4. The summary is based on Edwards book, pages 45-48.
  5. Nouv. Mém. Acad. Berlin, année 1771, 125; ibid. année 1773, 275; ibid année 1775, 351.
  6. A. David Christopher, A partition-theoretic proof of Fermat’s Two Squares Theorem”, Discrete Mathematics, 339 (2016) 1410–1411.
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