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Sei ν eine (komplexe) Zahl welche die Gleichung ν²+ν+1=0 erfüllt. Welchen Wert besitzt dann ν³ ? (bearbeitet)
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Für n,m∈ℕ\{1}, n≥7, m<n mit m|n (m teilt n), gilt: 1 + m + n/m < n
Beweis: Seien n,m∈ℕ\{1}, n≥7, m<n und m|n. Damit gilt: n/2 ≥ m ≥ 2. Sei f(m) := 1 + m + n/m. Dann ist f'(m) = 1 - n/m² = 0 ⇔m² - n = 0 ⇔|m| = √(n). Es gilt: f''(m) = 2n/m³ > 0 ∀n,m∈ℕ\{1}, also insbesondere für √(n). Damit hat f einen Tiefpunkt in √(n). Da f stetig ist und f' < 0 für 2 ≤ m < √(n)... (bearbeitet) mehr lesen
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