prove-that-p-n-is-integer-n-Z-p-n-3n-7-7n-3-11n-21-

Question Number 210666 by universe last updated on 15/Aug/24

prove that p (n) is integer \forall n \in Z

p (n) = \frac{{3 n}^{7} + {7 n}^{3} + 11 n}{21}

Answered by A5T last updated on 15/Aug/24

L e t f (n) = 3 n^{7} + 7 n^{3} + 11 n = n (3 n^{6} + 7 n^{2} + 11)

N o t e t h a t w h e n 7 ∣ n, t h e n 7 ∣ f (n)

S o, w h e n (7, n) = 1 \Rightarrow n^{6} \equiv 1 (m o d 7)

\Rightarrow f (n) \equiv n (3 + 11) = 14 n \equiv 0 (m o d 7) \Rightarrow 7 ∣ f (n)

S i m i l a r l y w h e n 3 ∣ n, t h e n 3 ∣ f (n)

w h e n (3, n) = 1 \Rightarrow n^{2} \equiv 1 (m o d 7) \Rightarrow n^{6} \equiv 1 (m o d 7)

\Rightarrow f (n) \equiv n (3 + 7 + 11) = 21 n \equiv 0 (m o d 3) \Rightarrow 3 ∣ f (n)

\Rightarrow 21 ∣ f (n) \Rightarrow p (n) = \frac{f (n)}{21} \in Z

Commented by universe last updated on 15/Aug/24

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