From 836bb2966761354800eee7af6b3b86491c5a848a Mon Sep 17 00:00:00 2001 From: Jim Hefferon Date: Sat, 28 Jul 2018 10:12:39 -0400 Subject: [PATCH] up to the fundamental thm --- ibl.tex | 210 ++++++++++++++++--------------- output/ibl-compact-maxlength.pdf | Bin 212218 -> 212151 bytes output/ibl-maxlength.pdf | Bin 251178 -> 251093 bytes 3 files changed, 110 insertions(+), 100 deletions(-) diff --git a/ibl.tex b/ibl.tex index 5b017fe..635ff3c 100644 --- a/ibl.tex +++ b/ibl.tex @@ -1726,35 +1726,36 @@ Verify each. \begin{problem} Prove. \begin{exes} \begin{exercise} - A number $n>1$ is composite if and only if it can be decomposed into + A number $n$ is composite if and only if it can be decomposed into the product of two factors $n=a\cdot b$ - such that $11$ has a prime divisor~$p$ + Every composite number~$n$ has a prime divisor~$p$ with $p\leq \sqrt{n}$. This inequality cannot be made strict. \end{exercise} \begin{answer} - Because~$n$ is greater than~$1$ and - composite there are numbers~$a,b\in\Z$ with - $1\sqrt{n}$ and~$b>\sqrt{n}$ then we would have that - $a\cdot b>\sqrt{n}\cdot\sqrt{n}=n$, - so at least one of these two factors must be less than or equal + Because~$n$ is + composite it decomposes into $n=ab$ for some integers~$a,b$ with + $1\sqrt{n}$ and~$b>\sqrt{n}$ then we would have the strict inequality + $a\cdot b>\sqrt{n}\cdot\sqrt{n}=n$. + Thus at least one of these two numbers must be less than or equal to~$\sqrt{n}$. - Let $c$ be a factor such that - $10$, -each of which is greater than~$1$. -Thus~$m$ has a prime factor~$p_i$. -But dividing~$m$ by any~$p_i$ -leaves a remainder of~$1$, which -is a contradiction. -Hence a finite -list of primes is an impossibility. +which is a product $2\cdot 3\cdots p_{n-1}$ of numbers +that are each greater than~$1$. +Because~$m$ is greater than~$1$, it has a prime factor~$p_i$. +But dividing~$m$ by any~$p_i$ leaves a remainder of~$1$. +That's a contradiction and +hence a finite +list of primes is impossibile. + +\smallskip \textit{Alternate proof.} For contradiction suppose there are finitely many primes -$2,3,\ldots,p_k$, let $n$ be the product $2\cdot 3\cdots p_k$, +$2,3,\ldots,p_k$, let $n$ be their product $2\cdot 3\cdots p_k$, and consider $n-1$. Since $n-1$ is greater than each prime (as $n$ is greater than $3\cdot\cdots p_k$ by a factor of $2$), @@ -1841,34 +1845,36 @@ But then $p_i$ divides $n-(n-1)=1$, which is impossible. \end{problem} \begin{problem} -Suppose that $p$ is a prime. Prove each.\label{ex:EuclidsOtherLemma} +Let $p$ be prime. Prove each.\label{ex:EuclidsOtherLemma} \begin{exes} \begin{exercise} If $p\divides ab$ then either $p\divides a$ or $p\divides b$. \end{exercise} \begin{answer} Let $d=\gcd(p,a)$. - Then $d$ is a positive number such that $d\divides p$ and~$d\divides a$. - Because $p$ is prime, either $d=p$ or~$d=1$. + Then $d$ is positive, and $d\divides p$, and~$d\divides a$. + Because $p$ is prime, its only divisor possibilities are $d=p$ and~$d=1$. If $d=p$ then $p\divides a$. If $d=1$ then $p$ and~$a$ are relatively prime, - so Euclid's Lemma~\ref{ex:EuclidsLemma} shows that - $p\divides b$. + so Euclid's Lemma, Exercise~\ref{ex:EuclidsLemma}, gives that $p\divides b$. \end{answer} \begin{exercise} - If $p\divides a_0\cdot a_1\cdots a_{n-1}$ then $p$ divides at least one~$a_i$. + If $p\divides a_0\cdot a_1\cdots a_{n-1}$ for $n\geq 2$ + then $p$ divides at least one~$a_i$. \end{exercise} \begin{answer} - We prove this by induction. + This argument uses induction on~$n$. The base step is $n=2$, that there are two numbers in the product, which is proved in the prior item. For the inductive step assume that the statement is true for $n=2$, \ldots, - $n=k$ and suppose that $n=k+1$, that $p\divides a_0\cdots a_{k-1}a_k$. - Taking $a_0\cdots a_{k-1}$ as~$a$ and $a_k$ as~$b$, the prior item applies - to show that $p\divides a_0\cdots a_{k-1}$ or~$p\divides a_k$. - In the latter case we are done, while in the former case the - inductive hypothesis applies to show that $p$ divides one of the factors. + $n=k$ and consider the $n=k+1$ case, $p\divides a_0\cdots a_{k-1}a_k$. + Take $a_0\cdots a_{k-1}$ as~$a$ and $a_k$ as~$b$ + so that the prior item applies. + The prior item's conclusion is that + $p\divides a_0\cdots a_{k-1}$ or~$p\divides a_k$. + In the latter case we are done while in the former case the + inductive hypothesis gives that $p$ divides one of the factors. \end{answer} \end{exes} \end{problem} @@ -1879,71 +1885,75 @@ Suppose that $p$ is a prime. Prove each.\label{ex:EuclidsOtherLemma} % http://gowers.wordpress.com/2011/11/18/proving-the-fundamental-theorem-of-arithmetic/ \begin{problem} \notetext{Fundamental Theorem of Arithmetic} Any number $n>1$ can be expressed as a product of primes - $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ and this expression is - unique:~if - $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ and the primes - are in ascending order $p_11$ can be written as a product of one or more primes. + Prove that any $n>1$ can be written as a product of primes. \end{exercise} \begin{answer} We do induction. For the base step, observe that $n=2$ is the product of one prime. 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