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φ(−n) = φ(n), φ(1) = 1, φ(0) = 0 - libzahl - big integer library |
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git clone git://git.suckless.org/libzahl |
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--- |
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commit 8092c767cb5f872b62a0cabbef793a08643497db |
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parent dd753f78b0c39d86a4cccca08996df303762e532 |
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Author: Mattias Andrée <[email protected]> |
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Date: Thu, 28 Jul 2016 22:55:43 +0200 |
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φ(−n) = φ(n), φ(1) = 1, φ(0) = 0 |
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Signed-off-by: Mattias Andrée <[email protected]> |
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Diffstat: |
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M doc/exercises.tex | 8 ++++++-- |
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1 file changed, 6 insertions(+), 2 deletions(-) |
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--- |
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diff --git a/doc/exercises.tex b/doc/exercises.tex |
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@@ -262,10 +262,13 @@ which calculates the totient of $n$. Its |
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formula is |
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\( \displaystyle{ |
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- \varphi(n) = n \prod_{p \in \textbf{P} : p | n} |
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+ \varphi(n) = |n| \prod_{p \in \textbf{P} : p | n} |
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\left ( 1 - \frac{1}{p} \right ). |
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}\) |
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+Note that, $\varphi(-n) = \varphi(n)$, $\varphi(0) = 0$, |
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+and $\varphi(1) = 1$. |
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+ |
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\end{enumerate} |
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@@ -671,7 +674,8 @@ So, if we set $a = n$ and $b = 1$, then we iterate |
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of all integers $p$, $2 \le p \le n$. For which $p$ |
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that is prime, we set $a \gets a \cdot (p - 1)$ and |
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$b \gets b \cdot p$. After the iteration, $b | a$, |
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-and $\varphi(n) = \frac{a}{b}$. |
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+and $\varphi(n) = \frac{a}{b}$. However, if $n < 0$, |
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+then, $\varphi(n) = \varphi|n|$. |
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