From 917abf0d416551e70bb786f8de0da898b7053d25 Mon Sep 17 00:00:00 2001 From: "Robin P. Clark" Date: Tue, 16 Jun 2015 15:09:24 +0100 Subject: [PATCH] rime=i+i+3; --- papers/JOURNAL_fmea_sw_hw/sw_hw_hierarchy.dia | Bin 1849 -> 1890 bytes papers/fermat/fermat.tex | 206 ++++++++++++++++++ 2 files changed, 206 insertions(+) create mode 100644 papers/fermat/fermat.tex diff --git a/papers/JOURNAL_fmea_sw_hw/sw_hw_hierarchy.dia b/papers/JOURNAL_fmea_sw_hw/sw_hw_hierarchy.dia index 5a110af32983f63ed57dbb66ce91a2bb4fcfcd35..96a3399ad0a9c31a6e019e84b0a1992440ecd761 100644 GIT binary patch literal 1890 zcmV-o2c7sIiwFP!000021MOT*lbbjYz57?FoWmwnL0ZuB6CT8MNO#``!hH&Hne1am?NOa}9~XS%j)JF&c=U99`Z%XGi{bPD#~ zyuGD0l)W94hk#2a+3E5yTZMtlUNEzIn*%+~t7fB@W_`Eau@h5X+7aslA`*`>t-AlX zk!e&t;EHjO@Kt#Kbhb4o$>eY8t7@=cm52k82IwvNnEV)SVi1Tx!r*53hgqMj-QhuJ zz=M0UwM;f9BskwbvEwJV?oXyo?FphZ*%XH5Jn;5$^F#{!m-~VO9QNwe}ASdxnc0rqe@843n|3Wn-5RW^nA7 zM&)*30=KcwCG-ESa?`ecLLlfgKu>ilcZoqU=MEiPX8xNP z7Q;5*td9M>;{6#OKrY@$C?wEy6+=pr#3CAiRsFR{f>k8aUr;3PWz@^NZh3Do*IwSc zC+~J!+VU{T#n#ctS{}AaK?iZC*fEF=3a?Y#+l}cMZKpP-wzqZY)ONei%G+r>YwIKw zY}X*>*=tEhxycgZ#N)MpgMDRUPO|>TjiQNYnxUdvF6@lOh>M{+ckf6 zG^#t@$=rK&KT+M?)|tD7O&848!mifjeGOwlBuqSPdZak@I>6C_Vmv@N+@gkyUG7k8TtPDKasKnuR8L3R>1fXc zdn&pr6|E;Kj%Bfgp6&V8(<^8Wj8xgCN{Rv=w0ZSr$oDIeb~?RC8slB2Y=~pZ^i~c5 zLY~CXBEh&q;Sg_!@^T(yd9_1h9s_Y~ZAH|heI9dK?T~qaM1~q3wsog56zrDsF*PxR z)ZfDnaXXN}d|H3th$d0uf6q`dMOjr))Q1gM$YS=4vkx4)wly^|dTNgOyJi+pv1X?SF?0iQqD++~pfd{#UUO8_$ ztfTatUNJMbFsZQw=`-^a_1w9Dt9Wi5rA{m2!PkHEW3{YPKyBe1&UFeJh$f9 zBzo&nN*otLfc(RD*DF!2PGt1IM{+IuAm^2~qcC2s)F@dg_|?n1c6o0$)n4B_SUYUB zucC7>ghUlcNH3~6{kKcK|5(%ds>%u;Vk0O)EU+-o_522bP0a=M33Y8}+B8*n4 z%u^Ye^$ez+!J2Hg=dTX(7cnoe`GFNnB3k!huTe<8O{+YC^tcn7w0b_e!tRHC8|?Y$ cKjowHy!c0`w}aY$O1&NY3(wFB8ckgQ04NKn-2eap literal 1849 zcmV-92gdjxiwFP!000021MQsObJ{o%$KUxYH1y%x8CTenWy_^W=X!IUzI5)Pz2}@T z0(3PtV~eDDxxf9^KLQv`0vK^+&UBi@M$y-jclV%9N)V;-SVSc!M=f0~Lpg5QX<{OG1X(|Yl4FnD-)z|mtSbdusoEpa9X z|A;6O186kp-}VsN>|iXk$j?=0h1RJWEwx0kn9G~~NQB>~X|jyR{jycnE=-~%MfW1Q z>A#v3ul}IwW>D+rnZ6fdDo3dl-}iTI!z&0*>3Sj4?QZ9bBva5tKQ4Bf?CB@}y3wX; zl|iF;di&~E_f>gG)roaqO&7XXM9)P!Rq?J4%uQ5I1i_pT0+Yc49@xI?J5O_Ryxi!r za?xew(#y)J>~4{yIu%OqN{*5wk|Hiub-I+N>tvybz{H*#R*O+gl-9`$*FORGCyt-AlX z#x$yKa20Wo2(`L@I@*|%Wc0TTbv4+pTEt_Kj?r86G5OJ7uR&l1qQ*DFZur>O{b;n^dkWVt;%OwG6fBAZHlzf*mIFbaohBwDIll{& zG~QhvoNGLVL=i=H$Fzzd6jOWi<+RY6Nj076{bC&RnwIUKCN@gau}tskZrk9vj>!L= zrjgB(hr5P#>C4b<4iJO2Hyq#|-{cRy;U7-rA5LrkaByJx*kdL=q|~q(8#*rb31Jq; zo=r~F)cI7rfP<|z`)+Bmey+9KT55kyCi+37vbSf&{S2n|t$C7IH!0@l(L~Hu^avLw z;;fG{{g}s7h=t{^ewNX_)G8F4VgL9m*m24R$F#Q{({|g5)~)bBOFTtrI|dvS2Tl=! zh4D0=%jUZ&R#hiO|=-Byp zVmS=Ee7!sN^TPWx+<+Y3NN6O`WEn$BlEfnFfmQuAPsYngq`#m@-pi=NyJ@^Pm}`gk z>%+U-Hf#l$N@;6SEw+=FRM0Qd`$ z>I_9H(;eMS)4j=HJG@^X-o5tw?H1$-)MW0?iGWU+|N7CJIow+;M_Q7l&>>EwqS27N)gh_zyfE0(pRcyVlvUinZeIi0Qi?YE( z2Xo`8dZT@IIRBY(z9&}k*0G9V-{M9$<3YmX4z+yjbB{X1N|Nz_^PfLVb>YZ(INCA6 zE)-of6s;yIj%Bfhp6&S7(>rJe5vj6GofHK;X#MU@pYK*8ZFPEfra32HBcc?_&$YdfL=?edu8YKQDIBr?g2McuOD0$I$Cadr!b>)VoaTGu!dh#MBe#AJ{v ziij9?*pND)=C|4|d=cjDMsyI$Z5-6L38_Jp6KvN}&O4MJhw|1{cZ6VIR7G4{Q3=oTYKU`b|f1;}mbS%Z}q$$nn+{j}E&)WiVS; z22&qRt!U zRhTCE)}&68D3WPMa^oa#w#yFXSBUb~1&(=~VEK~-Mk|Hf{E1vv6bkV|4{{B&IBz+u zgLqCan3+46)S?9GGV>#O?w!F^0>7|bo0)Z%LWY3_gTH{M6xBi=7{E1D3$35-K{5&f z>ahwlR6OJ(GYe%3-pSA+l?#!|F`9tIEz%^ESq4Ap1cAwqyETCEXM7uNxmyBU;S#_< z!*jkKn1YWTvk&@uxrPlPEhzK?!%z9P8q-1i_b$qRUej{ij{nE;pP!)=#mh;|u}#cY zqm(*6c4_`myC0OQ&Q;{|zejRKeZYCecBJCPQV)`)hHo9-jpM!9R6D+3Vdb#XzJ|`j z6c&2~7-BMPtyG9zeE81ueMmSA*fr!WA4MinxO$cmJt#dx0*=vfGyI24KNEofj diff --git a/papers/fermat/fermat.tex b/papers/fermat/fermat.tex new file mode 100644 index 0000000..71f3728 --- /dev/null +++ b/papers/fermat/fermat.tex @@ -0,0 +1,206 @@ + + +%%% OUTLINE + + + + +%\documentclass[twocolumn]{article} +\documentclass{article} +%\documentclass[twocolumn,10pt]{report} +\usepackage{graphicx} +\usepackage{fancyhdr} +%\usepackage{wassysym} +\usepackage{tikz} +\usepackage{amsfonts,amsmath,amsthm} +\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows} +%\input{../style} +\usepackage{ifthen} +\usepackage{lastpage} + +\def\layersep{1.8cm} + +\linespread{1.0} + +\begin{document} + % numbers at outer edges +\pagenumbering{arabic} % Arabic page numbers hereafter +\author{R.Clark$^\star$, \\ + $^\star${\em Energy Technology Control, UK. r.clark@energytechnologycontrol.com} \and $^\dagger${\em University of Brighton, UK} +} + +%\title{Developing a rigorous bottom-up modular static failure mode modelling methodology} +\title{fermat} +%\nodate +\maketitle + +\today + +\paragraph{Keywords:} fermat; prime; +%\small + +\abstract{ % \em +} % abstract + + + +\section{Introduction} +Fermat's Last Theorem +states that no three positive integers a, b, and c can satisfy the +equation $a^n + b^n = c^n$ for any integer value of n greater than two. + + + +\section{Breaking these positive integers into constituent primes} + +Any positive integer can be represented as a collection (or bag) of prime numbers multiple together. +A function $bpf()$ or `bag of prime factors' is defined to represent this. +\begin{equation} + \prod{bpf(a)}^n + \prod{bpf(b)}^n = c^n +\end{equation} + +The function $bpf()$ will always contain 1. + +The numbers $a$ and $b$ may have common and will have uncommon prime factors; these can be collected into +three bags, those only in a $ubpf(a)$, those only in b, $ubpf(b)$ and those common, $cbpf(a,b)$. + +\begin{equation} +\label{eqn:primesexpanded0} + 2 \prod{cbpf(a,b)}^n \prod{ubpf(a)^n} + \prod{cbpf(a,b)}^n \prod{ubpf(b)^n} = c^n +\end{equation} + +this can be re-written as + +\begin{equation} +\label{eqn:primesexpanded1} + 2 \prod{cbpf(a,b)}^n \big( \prod{ubpf(a)^n} + \prod{ubpf(b)^n} \big) = c^n +\end{equation} + +These are all prime numbers, and although some may be repeated within their bags +a prime number can only exist in one of the bags. + +Also all these prime numbers are greater than two and therefore odd. + +So this becomes a product of a list of prime numbers in ${cbpf(a,b)}$. +The common prime factors between a and b multiplied +by the uncommon prime numbers. +Let $\prod{ubpf(a)^n} + \prod{ubpf(b)^n = k$. + +\begin{equation} +\label{eqn:primesexpanded2} + 2 \prod{cbpf(a,b)}^n k = c^n +\end{equation} + + + + +Adding two prime numbers at any power greater than 1 +and then taking a root means getting an irrational number. + + + +% +% \begin{equation} +% \label{eqn:primesexpanded} +% \prod{cbpf(a,b)}^n \big( \prod{ubpf(a)^n} + \prod{ubpf(b)^n} \big) = c^n +% \end{equation} +% +% \begin{equation} +% a^n + b^n = \prod{bpf(c)^n} +% \end{equation} +% +% +% %assuming its true $c^n$ must be $ 2 \prod{cpf(a,b)}^n \prod{upf(a)^n} \prod{upf(b)^n} $ +% +% +% % +% % \begin{equation} +% % 2 \prod{cpf(a,b)}^n = \frac{\prod{bpf(c)^n}}{\prod{upf(a)^n} \prod{upf(b)^n}} +% % \end{equation} + +\section{conditions for having a integer root} + +To have an integer root $n$ all prime numbers that comprise the number to be rooted must be at least +to the power of $n$. +Consider the square root of 144. +This can be written as +$12 \times 12$ or breaking it down into prime numbers +$2 \times 2 \times 3 \times 2 \times \times 2 \times 3$ or $ 2^4 \times 3^2 $. +Taking the square root means halving the powers $ \sqrt{2^4 \times 3^2} = 2^2 \times 3$. + +To get an nth root you need all the prime numbers that comprise +that number to be at the power of n or greater. + +That means that ${ubpf(a)^n}$ and ${ubpf(b)^n}$ multiply the prime numbers in $cbpf(a,b)$ +$n$ times each. +These are a component of $c^n$. + + +\begin{equation} +\label{eqn:primesexpanded1} + 2 \prod{cbpf(a,b)}^n = \frac{c^n}{\big( \prod{ubpf(a)^n} + \prod{ubpf(b)^n} \big) } +\end{equation} + +% +% \begin{equation} +% \label{eqn:primesexpanded1} +% \prod{ubpf(a)^n} + \prod{ubpf(b)^n} = \frac{c^n}{ \prod{cbpf(a,b)}^n } +% \end{equation} +% +% +% $c^n$ must contain $ \prod{cbpf(a,b)}^n $ +% %Try to find a and b such that a^2 + b^2 = 144; +% +% +% \begin{equation} +% \label{eqn:primesexpanded1} +% \prod{ubpf(a)^n} + \prod{ubpf(b)^n} = \frac{c^n}{ \prod{cbpf(a,b)}^n } +% \end{equation} +% + +%It should be even because its multiplied by 2. +% It must have all the common factors of $a$ and $b$ twice but the uncommon factors only once. +% This seems to be an apparent contradiction. +% It means the $2 \prod{cpf(a,b)}^n $ term is multiplied by at least one other prime number. % and therefore cannot have an nth root. +% A number must consist of n times of all its prime number can give an integer nth root. +% Because a and b are different they must consist of at least one difference in prime numbers. +% +% Taking equation~\ref{eqn:primesexpanded} +% and re-writing: +% \begin{equation} +% \label{eqn:primesexpanded2} +% \sqrt[n]{2}^n \prod{cbpf(a,b)}^n \prod{ubpf(a)^n} \prod{ubpf(b)^n} = c^n +% \end{equation} +% +% +% Taking the nth root of both sides of equation~\ref{qn:primesexpanded2} gives +% +% \begin{equation} +% \label{eqn:primesexpanded2} +% \sqrt[n]{2} \prod{cbpf(a,b)} (\prod{ubpf(a)} \prod{ubpf(b)}) = c +% \end{equation} +% + + + +Which means that a product of $c$ is a root of 2, it is therefore irrational +and not a whole number. + + +If $c$ is even 2 can be divided from each side until only +both $c$ and $ \prod{cbpf(a,b)} \prod{ubpf(a)} \prod{ubpf(b)} $ +are odd. The $\sqrt[n]{2}$ term remains. The result $c$ is therefore irrational. + +%Adding $a^n$ and $b^n$ where a and b are different means adding primes to th power of N +%which means they have no integer nth root. + +{ +\footnotesize +\bibliographystyle{plain} +\bibliography{../../vmgbibliography,../../mybib} +} + +\today +%\today +\end{document} +