Exercise 0.2.5 (Some values of $\varphi$)

Question

Determine the value $\varphi(n)$ for each integer $n \leq 30$ where $\varphi$ denotes the Euler $\varphi$-function.

Richard Ganaye · Accepted Answer

\begin{proof} 
We know that, for $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_s^{\alpha_s}$ (where $p_1,p_2,\ldots,p_s$ are distinct primes), then
$$\varphi(n) = p_1^{\alpha_1 - 1}(p_1-1) p_2^{\alpha_2 - 1}(p_2-1)\cdots p_s^{\alpha_s - 1}(p_s-1).$$
For instance, for $n = 12 = 2^2 \cdot 3$,
$$\varphi(12) = 2^1 (2 - 1) 3^0 (3-1) = 4.$$
With Sagemath:
\begin{verbatim}
sage: for n in range (1,31):
....:     print(n, euler_phi(n))
....:     
(1, 1)
(2, 1)
(3, 2)
(4, 2)
(5, 4)
(6, 2)
(7, 6)
(8, 4)
(9, 6)
(10, 4)
(11, 10)
(12, 4)
(13, 12)
(14, 6)
(15, 8)
(16, 8)
(17, 16)
(18, 6)
(19, 18)
(20, 8)
(21, 12)
(22, 10)
(23, 22)
(24, 8)
(25, 20)
(26, 12)
(27, 18)
(28, 12)
(29, 28)
(30, 8)
\end{verbatim}

\end{proof}

Exercise 0.2.5 (Some values of $\varphi$)

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Exercise 0.2.5 (Some values of $\varphi$)

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