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**Example text**

Une définition arithmétique de “numeri idonei” a été donnée par Euler (cf. [3], vol. 1, p. 361 ; pour les corrections voir [4]). Démonstration. Le nombre N3 (n) des réprésentations propres du nombre n > 3 par la forme x 2 + y 2 + z2 est donné par la formule (cf. [3], vol. 2, p. 265) ⎧ ⎪ ⎨12h(−4n) pour n ≡ 1, 2, 5, 6 (mod 8) (1) N3 (n) = 24h(−4n) pour n ≡ 3 (mod 8) ⎪ ⎩ 0 pour n ≡ 0, 4, 7 (mod 8), où h(d) est le nombre des classes de formes binaires au discriminant d. Mais, d’après le théorème sur la duplication, on a : ⎧ λ−2 pour d ≡ 4 (mod 16) ⎪ ⎨2 (2) h(d) = p(d) · 2λ pour d ≡ 0 (mod 32) ⎪ ⎩ λ−1 2 pour d’autres cas, où λ est le nombre des facteurs premiers du nombre d.

K−1 , ηk+1 , . . , ηn ]) and σ (k) is a permutation of {1, 2, . . , n} such that Aσ (k) = Ak for all k. Then we have η ϑ xp p (12) xsϑs = (p) yσ σ(p) η (s) yσ σ(s) . Proof. By Theorem 1, equation (1) is equivalent to n Ak Xkmk = 0, (5) k=1 and equation (9) is equivalent to the equation n Ak Yklk = 0. (13) k=1 By reflexivity, symmetry and transitivity of the equivalence considered, the sufficiency of condition (11) is obvious. By formula (6), formula (12) is also obvious. To show the necessity of condition (11), suppose that equations (1) and (9) are equivalent.

Interscience, New York and London 1962. [7] S. Lubelski, Zur Reduzibilität von Polynomen in der Kongruenztheorie. Acta Arith. 1 (1936), 169–183, and 2 (1938), 242–261. [8] G. Pólya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, II. Springer, Berlin 1954. Originally published in Commentarii. Pontificia Academia Scientiarum II:20 (1969), 1–9 Andrzej Schinzel Selecta An improvement of Runge’s theorem on Diophantine equations Summarium. Auctor investigat quando aequatio cum duabus variabilibus infinitum solutionum integralium numerum habere possit.