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