Abstract
Which famous mathematical theorem has been proved most often? Pythagoras would certainly be a good candidate or the fundamental theorem of algebra, but the champion is without doubt the law of quadratic reciprocity in number theory. In an admirable monograph Franz Lemmermeyer lists as of the year 2000 no fewer than 196 proofs. Many of them are of course only slight variations of others, but the array of different ideas is still impressive, as is the list of contributors. Carl Friedrich Gauss gave the first complete proof in 1801 and followed up with seven more. A little later Ferdinand Gotthold Eisenstein added five more — and the ongoing list of provers reads like a Who is Who of mathematics.
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References
A. Baker: A Concise Introduction to the Theory of Numbers, Cambridge University Press, Cambridge 1984.
F. G. Eisenstein: Geometrischer Beweis des Fundamentaltheorems für die quadratischen Reste, J. Reine Angewandte Mathematik 28 (1844), 186-191.
C. F. Gauss: Theorema arithmetici demonstratio nova, Comment. Soc. regiae sci. Göttingen XVI (1808), 69; Werke II, 1-8 (contains the 3rd proof).
C. F. Gauss: Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et amplicationes novae (1818), Werke II, 47-64 (contains the 6th proof).
F. Lemmermeyer: Reciprocity Laws, Springer-Verlag, Berlin 2000.
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Aigner, M., Ziegler, G.M. (2010). The law of quadratic reciprocity. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00856-6_5
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