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Antti H S Laaksonen 2017-02-18 21:17:54 +02:00
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@ -105,7 +105,7 @@
\include{luku28} \include{luku28}
\include{luku29} \include{luku29}
\include{luku30} \include{luku30}
\include{kirj} %\include{kirj}
\cleardoublepage \cleardoublepage
\printindex \printindex

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luku21.tex
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@ -73,7 +73,7 @@ The factors are
The \key{sum of factors} of $n$ is The \key{sum of factors} of $n$ is
\[\sigma(n)=\prod_{i=1}^k (1+p_i+\ldots+p_i^{\alpha_i}) = \prod_{i=1}^k \frac{p_i^{a_i+1}-1}{p_i-1},\] \[\sigma(n)=\prod_{i=1}^k (1+p_i+\ldots+p_i^{\alpha_i}) = \prod_{i=1}^k \frac{p_i^{a_i+1}-1}{p_i-1},\]
where the latter formula is based on the geometric sum formula. where the latter formula is based on the geometric progression formula.
For example, the sum of factors of the number 84 is For example, the sum of factors of the number 84 is
\[\sigma(84)=\frac{2^3-1}{2-1} \cdot \frac{3^2-1}{3-1} \cdot \frac{7^2-1}{7-1} = 7 \cdot 4 \cdot 8 = 224.\] \[\sigma(84)=\frac{2^3-1}{2-1} \cdot \frac{3^2-1}{3-1} \cdot \frac{7^2-1}{7-1} = 7 \cdot 4 \cdot 8 = 224.\]
@ -89,7 +89,7 @@ and the product of the factors is $\mu(84)=84^6=351298031616$.
\index{perfect number} \index{perfect number}
A number $n$ is \key{perfect} if $n=\sigma(n)-n$, A number $n$ is \key{perfect} if $n=\sigma(n)-n$,
i.e., $n$ equals the sum of its divisors i.e., $n$ equals the sum of its factors
between $1$ and $n-1$. between $1$ and $n-1$.
For example, the number 28 is perfect, For example, the number 28 is perfect,
because $28=1+2+4+7+14$. because $28=1+2+4+7+14$.
@ -355,10 +355,10 @@ For example,
Numbers $a$ and $b$ are \key{coprime} Numbers $a$ and $b$ are \key{coprime}
if $\textrm{gcd}(a,b)=1$. if $\textrm{gcd}(a,b)=1$.
\key{Euler's totient function} $\varphi(n)$ \key{Euler's totient function} $\varphi(n)$
returns the number of coprime numbers to $n$ gives the number of coprime numbers to $n$
between $1$ and $n$. between $1$ and $n$.
For example, $\varphi(12)=4$, For example, $\varphi(12)=4$,
because the 1, 5, 7 and 11 because 1, 5, 7 and 11
are coprime to 12. are coprime to 12.
The value of $\varphi(n)$ can be calculated The value of $\varphi(n)$ can be calculated
@ -517,9 +517,9 @@ cout << x*x << "\n"; // 2537071545
\index{Diophantine equation} \index{Diophantine equation}
A \key{Diophantine equation} is of the form A \key{Diophantine equation} is an equation of the form
\[ ax + by = c, \] \[ ax + by = c, \]
where $a$, $b$ and $c$ are constants, where $a$, $b$ and $c$ are constants
and we should find the values of $x$ and $y$. and we should find the values of $x$ and $y$.
Each number in the equation has to be an integer. Each number in the equation has to be an integer.
For example, one solution for the equation For example, one solution for the equation
@ -575,10 +575,10 @@ and by multiplying this by 4, the result is
\[ \[
39 \cdot 8 + 15 \cdot (-20) = 12, 39 \cdot 8 + 15 \cdot (-20) = 12,
\] \]
so a solution for the equation is so a solution to the equation is
$x=8$ and $y=-20$. $x=8$ and $y=-20$.
A solution for a Diophantine equation is not unique, A solution to a Diophantine equation is not unique,
but we can form an infinite number of solutions but we can form an infinite number of solutions
if we know one solution. if we know one solution.
If a pair $(x,y)$ is a solution, then also all pairs If a pair $(x,y)$ is a solution, then also all pairs
@ -603,7 +603,7 @@ where all pairs of $m_1,m_2,\ldots,m_n$ are coprime.
Let $x^{-1}_m$ be the inverse of $x$ modulo $m$, and Let $x^{-1}_m$ be the inverse of $x$ modulo $m$, and
\[ X_k = \frac{m_1 m_2 \cdots m_n}{m_k}.\] \[ X_k = \frac{m_1 m_2 \cdots m_n}{m_k}.\]
Using this notation, a solution for the equations is Using this notation, a solution to the equations is
\[x = a_1 X_1 {X_1}^{-1}_{m_1} + a_2 X_2 {X_2}^{-1}_{m_2} + \cdots + a_n X_n {X_n}^{-1}_{m_n}.\] \[x = a_1 X_1 {X_1}^{-1}_{m_1} + a_2 X_2 {X_2}^{-1}_{m_2} + \cdots + a_n X_n {X_n}^{-1}_{m_n}.\]
In this solution, it holds for each number In this solution, it holds for each number
$k=1,2,\ldots,n$ that $k=1,2,\ldots,n$ that