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By Eiichi Bannai

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Bibliographie [1] C. H ERMITE : Sur la fonction exponentielle, Comptes rendus de l’Académie des Sciences (Paris) 77 (1873), 18-24 ; Œuvres de Charles Hermite, Vol. III, Gauthier-Villars, Paris 1912, pp. 150-181. [2] Y. I WAMOTO : A proof that π 2 is irrational, J. Osaka Institute of Science and Technology 1 (1949), 147-148. [3] J. F. KOKSMA : On Niven’s proof that π is irrational, Nieuw Archief voor Wiskunde (2) 23 (1949), 39. [4] J. , Journal de Mathématiques Pures et Appl. (1) 5 (1840), 192 ; Addition, 193-194.

Monthly 106 (1999), 652-665. [2] A. G RANVILLE & G. M ARTIN : Prime number races, American Math. Monthly 113 (2006), 1-33. [3] D. R. H EATH -B ROWN : Fermat’s two squares theorem, Invariant (1984), 2-5. [4] I. N IVEN & H. S. Z UCKERMAN : An Introduction to the Theory of Numbers, Fifth edition, Wiley, New York 1972. [5] H. R IESEL : Prime Numbers and Computer Methods for Factorization, Second edition, Progress in Mathematics 126, Birkhäuser, Boston MA 1994. [6] M. RUBINSTEIN & P. S ARNAK : Chebyshev’s bias, Experimental Mathematics 3 (1994), 173-197.

On trouve mention de ce problème dans deux lettres de Carl Friedrich Gauss datant de 1844 (publiées en 1900 dans le recueil des œuvres de Gauss). 5 d’Euclide selon lequel des pyramides de même base et de même hauteur ont le même volume. On obtiendrait ainsi une définition élémentaire du volume des polyèdres (ne reposant pas sur des arguments issus de l’analyse, et en particulier d’arguments de continuité). David Hilbert La croix est équicomplémentaire à un carré de même aire. En fait, la croix et le carré sont même équidécomposables.

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