\documentclass[% version=last,% paper=a5,% 12pt,% %headings=small,% toc=graduated,% default %toc=flat,% %toc=leveldown,% listof=leveldown,% %listof=nochaptergap,% bibliography=totoc,% %index=totoc,% twoside=yes,% reqno,% cleardoublepage=empty,% titlepage=no,% open=any,% %draft=false,% draft=true,% DIV=12,% headinclude=false,% pagesize]% {scrbook} %\pagestyle{empty} \usepackage[polutonikogreek,english]{babel} \usepackage{gfsporson} \newcommand{\gr}[1]{\foreignlanguage{polutonikogreek}{\relscale{0.9}\textporson{#1}}} \usepackage{yfonts} \usepackage{relsize} % Here \smaller scales by 1/1.2; \relscale{X} scales by X \usepackage{moredefs,lips} \usepackage{verbatim} \usepackage{verse,longtable} \renewenvironment{quote}{\begin{list}{} {\relscale{.90}\setlength{\leftmargin}{0.05\textwidth} \setlength{\rightmargin}{\leftmargin}} \item\relax} {\end{list}} \renewenvironment{quotation}{\begin{list}{}% {\relscale{.90}\setlength{\leftmargin}{0.05\textwidth} \setlength{\rightmargin}{\leftmargin} \setlength{\listparindent}{1em} \setlength{\itemindent}{\listparindent} \setlength{\parsep}{\parskip}} \item\relax} {\end{list}} \usepackage{multicol} \newcommand{\rn}[2]{\makebox[4cm]{#1\dotfill#2}} % for the roman numerals \newcommand{\dict}[1]{\textbf{\sffamily#1} #1} % for "The Mutual Letter" \usepackage{url} \usepackage{textcomp} % supposedly useful with \oldstylenums \usepackage[neverdecrease]{paralist} \usepackage{hfoldsty} % this didn't work until I added missing % brackets to some of the files. \newcommand{\dpow}[2]{\item[#2.]\makebox[2em][r]{#1}} \newcommand{\dlog}[2]{\item[#1.]\makebox[2em][r]{#2}} \newcommand{\dpowq}[2]{#2.\makebox[2em][r]{#1}} \newcommand{\dlogq}[2]{#1.\makebox[2em][r]{#2}} \usepackage{pstricks,pst-plot,pst-math,pst-eucl} \usepackage{amsmath,amssymb,amsthm,upgreek} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\divides}{\mid} \newcommand{\ndivides}{\nmid} \newcommand{\units}[1]{#1{}^{\times}} \DeclareMathOperator{\order}{ord} \newcommand{\ord}[2]{\order_{#1}(#2)} \DeclareMathOperator{\lcm}{lcm} \newcommand{\size}[1]{\lvert#1\rvert} \newcommand{\logicrel}[1]{\;\mathrel{\text{#1}}\;} \newcommand{\euphi}{\upvarphi} \renewcommand{\leq}{\leqslant} \newtheorem*{theorem}{Theorem} \theoremstyle{definition} \newtheorem*{definition}{Definition} \setlength{\columnseprule}{0.5pt} \begin{document} \title{Discrete Logarithms} \subtitle{Mathematics and Art} \author{David Pierce} \date{December 17, 2017--April 4, 2018} \publishers{Matematics Department\\ Mimar Sinan Fine Arts University, Istanbul\\ \url{mat.msgsu.edu.tr/~dpierce/}\\ \url{polytropy.com}} \maketitle \thispagestyle{empty} \vfill \begin{center} \psset{unit=25mm} \begin{pspicture}(-1,-1.2)(1,1.2) %\psgrid \pscircle(0,0)1 \uput[u](1;90){15} \uput[67.5](1;67.5){11} \uput[55](1;45){16} \uput[22.5](1;22.5){14} \uput[r](1;0){8} \uput[-22.5](1;-22.5){7} \uput[-45](1;-45){4} \uput[-67.5](1;-67.5){12} \uput[d](1;-90){2} \uput[-112.5](1;-112.5){6} \uput[-135](1;-135){1} \uput[-157.5](1;-157.5){3} \uput[l](1;180){9} \uput[157.5](1;157.5){10} \uput[135](1;135){13} \uput[112.5](1;112.5){5} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \uput[d](1;90){16} \uput[-112.5](1;67.5){14} \uput[-135](1;45){8} \uput[-157.5](1;22.5){7} \uput[l](1;0){4} \uput[157.5](1;-22.5){12} \uput[135](1;-45){2} \uput[112.5](1;-67.5){6} \uput[u](1;-90){1} \uput[67.5](1;-112.5){3} \uput[45](1;-135){9} \uput[22.5](1;-157.5){10} \uput[r](1;180){13} \uput[-22.5](1;157.5){5} \uput[-45](1;135){15} \uput[-67.5](1;112.5){11} % \pscircle(0,0){0.8} \uput[d](0.75;90){8} \uput[-112.5](0.75;67.5){9} \uput[-135](0.75;45){10} \uput[-157.5](0.75;22.5){11} \uput[l](0.75;0){12} \uput[157.5](0.75;-22.5){13} \uput[135](0.75;-45){14} \uput[112.5](0.75;-67.5){15} \uput[u](0.75;-90){16} \uput[67.5](0.75;-112.5){1} \uput[45](0.75;-135){2} \uput[22.5](0.75;-157.5){3} \uput[r](0.75;180){4} \uput[-22.5](0.75;157.5){5} \uput[-45](0.75;135){6} \uput[-67.5](0.75;112.5){7} \end{pspicture} \end{center} \vfill \tableofcontents \vfill \listoftables \vfill \listoffigures \vfill \addchap{Introduction} The core of this document is the two tables, of logarithms and antilogarithms respectively, constituting Chapter \ref{ch:lists}. The numbers in the tables may appear to be random. However, you can check in specific cases that each table undoes the other: for example, since the first table gives 564 as the logarithm of 319, the second table inevitably gives 319 as the antilogarithm of 564. The antilogarithm of 1 is 7, and each successive antilogarithm is either 7 times the previous, or else it is the remainder of that multiple after division by 997. Therefore the logarithms can be used as Briggsian or common logarithms once were, for computing products by taking sums. The logarithm of the product is the sum of the logarithms, though sums now are taken \emph{modulo} 996, and products \emph{modulo} 997. In the terminology of Euler, 7 is a \emph{primitive root} of 997. Chapter \ref{ch:math} reviews the mathematics, from Euclid to Gauss and beyond. If sufficiently interested, the layperson may follow the review, while the professional may still find something new. Anyone may contemplate the tables as conceptual art. I consider art as such in Chapter \ref{ch:art}, mainly through the work of R. G. Collingwood, but also Mary Midgley, Arthur Danto, and others. I review other examples of conceptual art. I quote theory and scholarship, poetry and fiction, mostly from books in my personal collection. The quotations may be considered as if they were readymades of Marcel Duchamp, or pictures in the exhibition that I am curating. \chapter{Art}\label{ch:art} \section{Creation} What counts as art today is broader than Collingwood contemplated in 1938 in \emph{The Principles of Art} \cite{Collingwood-PA}. Nonetheless, the book remains invaluable. In writing poems, or painting pictures, or composing quartets, or even---I would add---proving mathematical theorems, before you can employ a \emph{technique,} according to a \emph{plan,} you have to discover how to do everything in the first place. This need seems easily overlooked. Collingwood points it out. In creating your work of art, you cannot say---% you cannot \emph{express}---% in any precise way, what you are trying to do, before figuring out how to it. The figuring out is precisely the expressing of it. Expression is the key word. As Collingwood says on his page 151, \begin{quote} By creating for ourselves an imaginary experience or activity, we express our emotions; and this is what we call art. \end{quote} This is not a conclusion, but a halfway point; the text will end on page 336. It is important to read further, here into page 152: \begin{quote} What this formula means, we do not yet know. We can annotate it word by word; but only to forestall misunderstandings, thus. `Creating' refers to a productive activity which is not technical in character. `For ourselves' does not exclude `for others'; on the contrary, it seems to include that; at any rate in principle. `Imaginary' does not mean anything in the least like `make-believe', nor does it imply that what goes by that name is private to the person who imagines. The `experience or activity' seems not to be sensuous, and not to be in any way specialized: it is some kind of general activity in which the whole self is involved. `Expressing' emotions is certainly not the same thing as arousing them. There is emotion there before we express it\lips \end{quote} We are faced now with three problems: to understand (1) imagination, (2) emotion, and (3) their connection. \begin{quote} These problems must be dealt with\lips not by continuing to concentrate our attention on the special characteristics of aesthetic experience, but by broadening our view, so far as we can, until it covers the general characteristics of experience as a whole. \end{quote} I propose to consider this broadened view as encompassing mathematics. I say that art and mathematics are creations. You may disagree. In \emph{Heart and Mind} from 1981, in the chapter called Creation and Originality, Mary Midgley takes issue with the treatment of creation by Collingwood and others, especially Nietzsche and Sartre \cite[pp.\ 49--67]{Midgley-HM}. She begins with the importance of her subject, which is morality rather than art as such. \begin{quote} The creation of moral values is a pressing topic because, whether we use words like \emph{creation} or not, we all need to find new moral ideas to help us deal with a confused and changing world. The notion that these ideas must be totally new, that they should not rest at all on traditional supports, exists and concerns us all. \end{quote} The God of Genesis calls light into existence and \emph{then} sees that it is good \cite{KJV-Oxford}. God causes dry land to appear and \emph{then} sees it as good. Likewise with grass, herb, and tree, and with the lights of heaven, and so forth: first they are created, and then they are evaluated. Not even God just \emph{declares} what is good: its existence is by fiat, but not its goodness. As for ourselves, if we are no longer going to take our values from heaven, there is no sense in trying to do what not even its mythical ruler can do. This is what I understand Midgley to argue. ``If God is really dead,'' she says, ``why should we dress up in his clothes?'' We cannot just will things into existence, especially not goodness: \begin{quote}\sloppy The human will is not a mechanism for generating new thoughts out of nothing. It is a humble device for holding onto the thoughts which we have got and using them. \end{quote} The will then is not creative, but preservative. It may thus be humble, but it is still essential. Students need it, especially when they carry around the little electronic devices that are designed to draw their attention---to draw and quarter it, one might say. The student needs attention, application, \emph{persistence,} as I observed elsewhere \cite[p.\ 245]{MR3312989}. As expressing the thought, I quoted one of William Blake's ``Proverbs of Hell'' from \emph{The Marriage of Heaven and Hell} \cite[plate 7]{Blake}: \begin{quote} \centering If the fool would persist in his folly he would become wise. \end{quote} That the creativity of civilization depends on persistence is an argument for the \emph{rise} of what Julian Jaynes calls the bicameral mind, although the title of his 1976 book is \emph{The Origin of Consciousness in the Breakdown of the Bicameral Mind.} Other animals go about their business naturally, but civilization is an unnatural business. It requires us, in youth and later, to do things that we do not see the point of. The \emph{will} to do these things has needed to evolve. For Jaynes, one stage in this evolution was the hearing of voices that kept us at work. ``Let us consider a man,'' he says \cite[pp.\ 134 f.]{Jaynes}, \begin{quote} commanded by himself or his chief to set up a fish weir far upstream from a campsite. If he is not conscious, and cannot therefore narratize the situation and so hold his analog `I' in a spatialized time with its consequences fully imagined, how does he do it? It is only language, I think, that can keep him at this time-consuming all-afternoon work. A Middle Pleistocene man would forget what he was doing. But lingual man would have language to remind him, either repeated by himself, which would require a type of volition which I do not think he was then capable of, or, as seems more likely, by a repeated `internal' verbal hallucination telling him what to do\lips learned activities with no consummatory closure do need to be maintained by something outside of themselves. This is what verbal hallucinations would supply. \end{quote} How we have come to be where we are is indeed a puzzle, though I shall not dwell on Jaynes's attempt at a solution. We can think of the puzzle both on a ``special'' scale---% the scale of our species---% and on a personal scale, as Collingwood does in his last book, from 1942, \emph{The New Leviathan: Or Man, Society, Civilization, and Barbarism.} Here Collingwood takes issue with the notion of Rousseau that ``Man is born free, and everywhere he is in chains.'' ``I do not doubt,'' says Collingwood \cite[p.\ 176]{Collingwood-NL}, ``that truths, and important truths, can be told in Rousseau's language.'' However, \begin{quotation} \textbf{23.\ 93.} In human infancy the fact, as known to me at least, is that a man is born neither free nor in chains. \textbf{23.\ 94.} To be free is to have a will unhampered by external force, and a baby has none. \textbf{23.\ 95.} To be in chains is to have a will hampered by something which prevents it from expressing itself in action; and a baby has none. \textbf{23.\ 96.} A man is born a red and wrinkled lump of flesh having no will of its own at all, absolutely at the mercy of the parents by whose conspiracy he has been brought into existence. \textbf{23.\ 97.} That is what no science of human community, social or non-social, must ever forget. \end{quotation} I wonder whether Midgley forgets these facts in \emph{Heart and Mind.} She does recognize that creation can be perceived on a smaller scale than Genesis. Indeed, she quotes Collingwood from \emph{The Principles of Art} as showing this. Here he is, in an expansion of Midgley's quotation \cite[pp.\ 128 f.]{Collingwood-PA}. \begin{quotation}\noindent Readers suffering from theophobia will certainly by now have taken offence\lips Perhaps some day, with an eye on the Athanasian Creed, they will pluck up courage to excommunicate an arithmetician who uses the word three. Meanwhile, readers willing to understand words instead of shying at them will recollect that the word `create' is daily used in contexts that offer no valid ground for a fit of \emph{odium theologicum}\lips To create something means to make it non-technically, but yet consciously and voluntarily. Originally, \emph{creare} means to generate, or make offspring, for which we still use its compound `procreate'\lips The act of procreation is a voluntary act, and those who do it are responsible for what they are doing; but it is not done by any specialized form of skill\lips It is in this sense that we speak of creating a disturbance or a demand or a political system. The person who makes these things is acting voluntarily; he is acting responsibly; but he need not be acting in order to achieve any ulterior end; he need not be following a preconceived plan; and he is certainly not transforming anything that can properly be called a raw material. It is in the same sense that Christians asserted, and neo-Platonists denied, that God created the world. \end{quotation} Midgley objects, in a way that suggests to me that she has not really thought about what it means to grow up, or even what it means to compose an essay such as her own. \section{Gender} Midgley's experience of writing and life is no doubt different from mine. An important difference is connected to the English gendered pronouns. Our \emph{first}-person pronouns are epicene; but in the third person, I become he, while Midgley is she. The distinction is not imposed on us by nature. Each of us, including objects thought to be inanimate, is simply \emph o in Turkish, which is a language ``born free'' of ``the curse of grammatical gender'' \cite[II.26, p.\ 48]{Lewis2}. In English, we have a vestige of the curse, a vestige that can either reflect differences in experience, or \emph{effect} them. In 2013, Midgley wrote to \emph{The Guardian} as follows \cite{Midgley-Guardian}. She was responding to the question of ``why, though five quite well-known female philosophers emerged from Oxford soon after the war, few new ones are doing so today.'' \begin{quotation} As a survivor from the wartime group, I can only say: sorry, but the reason was indeed that there were fewer men about then. The trouble is not, of course, men as such---% men have done good enough philosophy in the past. What is wrong is a particular style of philosophising that results from encouraging a lot of clever young men to compete in winning arguments. These people then quickly build up a set of games out of simple oppositions and elaborate them until, in the end, nobody else can see what they are talking about. All this can go on until somebody from outside the circle finally explodes it\lips By contrast, in those wartime classes---which were small---men (conscientious objectors etc) were present as well as women, but they weren't keen on arguing. It was clear that we were all more interested in understanding this deeply puzzling world than in putting each other down. That was how Elizabeth Anscombe, Philippa Foot, Iris Murdoch, Mary Warnock and I, in our various ways, all came to think out alternatives to the brash, unreal style of philosophising---based essentially on logical positivism---% that was current at the time. \end{quotation} It is unfortunate that war had to create an opportunity, both for women to pursue and develop their thoughts, and for men to learn from them, as I have learned from Midgley. In \emph{Evolution As a Religion,} she rightfully critiques the presumption of some scientists (generally male) in making grand pronouncements on the meaning of life from physical theories. She quotes Steven Weinberg as saying, in an ``excellent and informative little book,'' \begin{quotation}\noindent The more the universe seems comprehensible, the more it also seems pointless. But\lips The effort to understand the universe is one of the very few things that lifts [\emph{sic}] human life a little above the level of farce, and gives it some of the grace of tragedy. \end{quotation} Midgley observes \cite[p.\ 87]{Midgley-ev}, \begin{quote} Since virtually the whole book has been devoted to expounding astrophysics, not to discussing it as an occupation, and certainly not to discussing other occupations with which it might compete, Weinberg's readers might find this an unexpected blow. They might feel rather shaken and degraded by the sudden revelation that their lives are probably valueless, and they might also ask the reasonable question: how does Weinberg know? \end{quote} Obviously Weinberg is only giving his opinion. The problem is not the rudeness of stating such an opinion, but the unscientific practice of deriving the opinion \emph{from} science, rather than recognizing it as connected with why one has done science in the first place. Here I may have passed to my own thought, only prompted by Midgley. Our subject was art and creation, and I still wonder whether Midgley has understood Collingwood when she says \cite[pp.\ 64 f.]{Midgley-HM}, \begin{quote} It may seem that at this point the word `create' has been diluted into complete triviality, that it simply means `make'. But it still keeps an awkward core of special meaning, and one that is important for Collingwood's theory of art. On his view, creators need not, indeed characteristically do not, know in advance what they are going to make. He sees the absence of a `preconceived end' as a mark of real art, a mark which distinguishes it from mere craft. But if you really do not know what you are trying to bring about, it is hard to see how you can do it, and harder still to see how you can be called responsible. Artists don't in fact often talk in this way. They are often quite willing to discuss their aims and problems. But whether or not sense can be made of this for art, in morals it is surely a non-starter. \end{quote} This shows the difficulty of understanding Collingwood. He does indeed distinguish art from craft; but there is no X-ray machine that you can feed artefacts into, and a light flashes green for art, red for craft. The same object has aspects of both. It is not even the \emph{physical} object that can be a work of art at all. We shall come back to this later, on page \pageref{physical}. After taking an examination, students want to know how they did. If they do not already know this, just from what they themselves have written on their papers, then they must not have had a preconceived end in any precise sense. They want a good grade, but they do not know what this really means. If they have done well, according to their teacher, they may still be proud, and they have some right to be, since they are responsible for what they did. I had an aim when I set out to write this essay. I could have talked about the aim in general terms. But the aim has grown, and grown precise, just as the essay has taken shape. In particular, at the beginning, I had no idea of the current sectional divisions of this essay. \section{Individualism} This essay is an \emph{expression.} The term was key for art of Collingwood's time, notably that of the \emph{Blaue Rieter} group, formed in Munich by Franz Marc in 1911. According to Herbert Read in \emph{A Concise History of Modern Painting} \cite[p.\ 228]{Read}, \begin{quote} \emph{Blaue Reiter} was the first coherent attempt to show that what matters in art---% what gives art its vitality and effect---% is not some principle of composition or some ideal of perfection, but a direct expression of feeling, the form corresponding to the feeling, as spontaneous as a gesture, but as enduring as a rock. \end{quote} Read begins his book with a long quotation from Collingwood's 1924 book, \emph{Speculum Mentis or The Map of Knowledge.} The idea is that, ``in art, a school once established normally deteriorates as it goes on'' \cite[p.\ 82]{Collingwood-SM}. Collingwood's ideas themselves continued to develop. He published \emph{Outlines of a Philosophy of Art} in 1925, but updated his views a dozen years later in \emph{The Principles of Art.} Concerning the quotation that Read makes, but does not really analyze, from \emph{Speculum Mentis,} I suggest that a school of art, once founded, declines, precisely because its very foundation constitutes the identification of a technique, and technique is not art. In its article on Aesthetics, the \emph{Internet Encyclopedia of Philosophy} \cite{iep-collingwood} is misleading to suggest that Collingwood ``took art to be a matter of self-expression.'' There was no need to add the restriction to the self. This assertion in the \emph{Encyclopedia} is indeed followed by the formula from \emph{The Principles of Art} quoted above, whereby art is a creating for our \emph{selves.} However, if one reads beyond the formula, also as above, then one sees how Collingwood was at pains to keep references to the self from being misunderstood. Creating art for ourselves includes doing it for others. One's imagination need not be private to oneself. The central lesson of \emph{mathematics} is that each of us has the right to decide, for her- or himself, what is true. Mathematical truth does not come down from heaven, but comes up from within each of us. It is like art in this way. Mathematical truth is nonetheless common. In mathematics, we have the responsibility of resolving disputes amicably, because anything on which there is fundamental disagreement is not mathematics. It may not be art either. What is \emph{liked} may differ from person to person, whether we are talking about art or mathematics. Some mathematicians do not like the method of proof by contradiction. They should still agree on whether a given proof by contradiction is \emph{correct} as a proof by contradiction. Likewise should we all be able to agree on whether something is art; but the truth of this assertion is not so clear as the corresponding one for mathematics. This is a \emph{practical} reason why everybody should learn some mathematics: it teaches the possibility, if not the obligation, of peaceful resolution of differences. The theme that what is \emph{mental} need not be merely \emph{personal} goes back to Collingwood's first book, \emph{Religion and Philosophy} of 1916 \cite[p.\ 93]{Collingwood-RP}. In the chapter called ``Matter,'' concerning this as distinguished from mind, Collingwood wrote, \begin{quote} A boot is more adequately described in terms of mind---by saying who made it and what he made it for---than in terms of matter. And in the case of all realities alike, it seems that the materialistic insistence on their objectivity is too strong; for it is not true that we are unable to alter or create facts, or even that we cannot affect the course of purely ``inanimate'' nature. Materialism, in short, is right as against those theories which make the world an illusion or a dream of my own individual mind; but while it is right to insist on objectivity, it goes too far in describing the objective world not only as something different from, and incapable of being created or destroyed by, my own mind, but as something different and aloof from mind in general. \end{quote} Again, though art be expression, it is not \emph{self}-expression as such. In \emph{The Principles of Art,} even before formulating the tentative definition of art that we have seen, Collingwood argues that art is not merely a private concern. Art is for the world, for civilization, even though civilization may not respect this \cite[pp.\ 33 f.]{Collingwood-PA}: \begin{quote} Here lies the peculiar tragedy of the artist's position in the modern world. He is heir to a tradition from which he has learnt what art should be; or at least, what it cannot be. He has heard its call and devoted himself to its service. And then, when the time comes for him to demand of society that it should support him in return for his devotion to a purpose which, after all, is not his private purpose but one among the purposes of modern civilization, he finds that his living is guaranteed only on condition that he renounces [\emph{sic}] his calling and uses [\emph{sic}] the art which he has acquired in a way which negates its fundamental nature, by turning journalist or advertisement artist or the like; a degradation far more frightful than the prostitution or enslavement of the mere body. \end{quote} It \emph{is} disappointing that, in closing this passage, Collingwood takes up the mind-body dualism that he refuted in \emph{Religion and Philosophy.} One might say, echoing him there, ``Prostitution is more adequately described in terms of mind---% by saying it compromises one's capacity to love and be loved---% than in terms of matter.'' Collingwood reiterates the universality of art at the end of \emph{The Principles of Art} \cite[p.\ 333]{Collingwood-PA}, where he observes first (writing before 1938) that English painting and literature aim no longer just to amuse the wealthy, but to be competent as art. \begin{quotation}\noindent But the question is whether this ideal of artistic competence is directed backwards into the blind alley of nineteenth-% century individualism, where the artist's only purpose was to express himself, or forwards into a new path where the artist, laying aside his individualistic pretensions, walks as the spokesman of his audience. In literature, those who chiefly matter have made the choice, and made it rightly. The credit for this belongs in the main to one great poet, who has set the example by taking as his theme in a long series of poems a subject that interests every one, the decay of our civilization. \end{quotation} The poet is T. S. Eliot. Collingwood's conclusion is preceded by theory. After the formula for art from his page 151 quoted earlier, in starting to develop a theory of the imagination, Collingwood distinguishes thought from feeling. One distinction is that while feelings are private, thoughts are potentially public, or held in common \cite[p.\ 157]{Collingwood-PA}. One's own feeling of cold has no relation to anybody else's; but the thought that a house is ten degrees Celsius is the same for everybody in the house who has the thought. \section{Eros} By bringing feelings into consciousness, art allows them to be shared. Art is ultimately identified with \emph{language.} This is not language as a system for communication: developing such a system requires language in the first place. We are talking about language such as Archimedes uttered, when he exclaimed ``Eureka!''\ in the story told by Vitruvius \cite[pp.\ 36 f.]{MR13:419b}. The expression of the mathematician was not just the first-person singular perfect form \gr{e<'urhka} of the verb \gr{eisoskel'hs -'es}, which combines \gr{>'isos -h -on} \emph{equal} with \gr{t'o sk'elos -ous} \emph{leg}. In Euclid's diagrams, a number is such a bounded straight line as is implicitly divisible into units, all being equal to one another or, in modern terms, having the same length. \section{Multiplication} It is possible to \textbf{multiply} one number, the \textbf{multiplicand,} by another number, the \textbf{multiplier.} This means to lay out the multiplicand as many times as there are units in the multiplier, so that a new number is obtained. The new number is the \textbf{multiple} of the multiplicand by the multiplier, and it is the \textbf{product} of the two numbers. To obtain a product, what we lay out is perhaps not strictly the multiplicand itself, but \emph{copies} of it, namely numbers that are equal to it. The distinction is lost in our notation. Five times six would appear to be, literally, six, laid out five times; this gives \begin{equation*} 6+6+6+6+6, \end{equation*} the sum we know as $30$. I propose to denote the product here as $6\cdot 5$, to be understood as six, multiplied by five. The multiplicand \textbf{measures} the product and is a \textbf{submultiple} of it; the multiplier \textbf{divides} the product. We can measure thirty apples by six apples: the result is five piles, each holding six apples. This means we can divide the thirty apples among five children: each child gets six apples. Without using this terminology, Alexandre Borovik discusses the distinction between measuring and dividing apples in \emph{Metamathematics of Elementary Mathematics} \cite{Borovik-Meta}. Using the results just discussed, how can we show that the thirty apples can \emph{also} be divided among six children? Why should the sum \begin{equation*} 5+5+5+5+5+5 \end{equation*} of six fives be equal to the sum of five sixes as above? We shall review Euclid's general proof of what we call the \emph{commutativity} of multiplication. The proof will involve \emph{ratios} of numbers. \section{The Euclidean Algorithm} Given a pair of numbers, we may transform it by subtracting the less from the greater. We can continue until the two numbers become equal. We call this process the \textbf{Euclidean Algorithm.} In the first two propositions of Book \textsc{vii} of the \emph{Elements} \cite{Euclid-homepage}, Euclid describes the process with the passive form of the verb \gr{>anjufair'ew}, \emph{to take away alternately.} It is a deficiency of the big Liddell--Scott--Jones lexicon \cite{LSJ} that Euclid is not cited under this word, from which can be derived the noun \emph{anthyphaeresis} (\gr{>anjufa'iresis}), meaning \emph{alternate subtraction.} At the end of the anthyphaeresis, either of the two equal numbers measures all of the numbers that came before, and so it is in particular a \textbf{common measure} of the original two numbers. Moreover, every common measure of these numbers measures every number found in the course of the anthyphaeresis; in particular, the common measure measures the last number, which is therefore the \textbf{greatest common measure} of the first two numbers. In the case where this greatest common measure is properly speaking not a number but a single unit, the two original numbers must be \textbf{prime to one another.} %\section{Proportion} I once considered teaching number theory on the pattern of Euclid, but then I found his approach too strange for the modern student. I did learn two things: (1) the implicit use of the Euclidean algorithm in the definition of proportion of numbers, and (2) the use of this definition in a rigorous proof of commutativity of multiplication. Suppose we apply the Euclidean Algorithm to two numbers, lying on the left and right respectively. At each step of the algorithm, we record first whether the left-hand or right-hand number is greater. Thus we may obtain a sequence of letters L and R. If this is the same as the sequence obtained from another pair of numbers, then, by Euclid's definition at the head of Book \textsc{vii}, the four numbers are \textbf{in proportion,} and the first two numbers have the \textbf{same ratio} as the second two numbers. Let us pass to modern symbolism in an example. If the first two numbers are $14$ and $10$, then the steps of the algorithm give us \begin{align*} &(14,10),& &(4,10),& &(4,6),& &(4,2),& &(2,2), \end{align*} whence $2$ is the greatest common measure of $14$ and $10$. From $21$ and $15$ we obtain \begin{align*} &(21,15),& &(6,15),& &(6,9),& &(6,3),& &(3,3), \end{align*} so $3$ is the greatest common measure of $21$ and $15$. In either case, the pattern of larger entries is LRRL, and therefore, by definition, \begin{equation}\label{eqn:prop} 14:10::21:15. \end{equation} This is not strictly an equation, but an \emph{identity.} The ratio $14:10$ is not \emph{equal} to $21:15$, but the two ratios are the \emph{same} as one another: they are one. Euclid's language makes the distinction between equality and sameness; the former is not used for ratios. If we repeat the last letter in LRRL, obtaining LRRLL, and if we replace subsequences of repeated letters with their numbers, we obtain the sequence $(1,2,2)$, whose entries appear in the continued fraction \begin{equation*} 1+\cfrac1{2+\cfrac12}. \end{equation*} This then is a way to represent the ratio $14:10$ or $21:15$. We may also note \begin{align*} &\begin{gathered} 14=2\cdot 7,\\ 10=2\cdot5, \end{gathered}& &\begin{gathered} 21=3\cdot7,\\ 15=3\cdot5, \end{gathered} \end{align*} where the repetition of the multipliers $7$ and $5$ is another way to verify the proportion \eqref{eqn:prop}. However, it is important that $7$ and $5$ are prime to one another, so that they are uniquely determined by either of the pairs $(14,10)$ and $(21,15)$. It will be a consequence of commutativity that \begin{equation}\label{eqn:cross} 14\cdot15=21\cdot10, \end{equation} that is, $2\cdot7\cdot3\cdot5=3\cdot7\cdot2\cdot5$. Nevertheless, in Euclidean mathematics, an equation like \eqref{eqn:cross} cannot serve as a \emph{definition} the proportion \eqref{eqn:prop}, simply because the equation does not immediately establish that something about the pair $(14,10)$ is the \emph{same} as for $(21,15)$. \section{Commutativity} Multiplication is certainly commutative in case one of the factors is unity; for the product then is simply the other factor. From the definition of proportionality, all ratios of the form $x:x\cdot a$ are the same. In saying this so compactly, we follow the convention established by Descartes \cite{Descartes-Geometrie}, whereby letters from the beginning of the alphabet denote constants, and from the end, variables. Since the ratio $1:1\cdot a$ is just $1:a$, we can conclude \begin{equation}\label{eqn:1:a::b:ba} 1:a::b:b\cdot a. \end{equation} Suppose now $a:b::c:d$, so that the steps of the Euclidean algorithm are the same, whether applied to $(a,b)$ or to $(c,d)$. These steps are then the same as for $(a+c,b+d)$, by what we call the commutativity of addition. For, assuming $a>b$, we must also have $c>d$, and so $a+c>b+d$, and consequently \begin{equation*} (a+c)-(b+d)=(a-b)+(c-d). \end{equation*} We conclude \begin{equation}\label{eqn:implies} a:b::c:d\logicrel{implies}a:b::a+c:b+d. \end{equation} As a special case, since $a:b::a:b$, we have $a:b::a\cdot2:b\cdot2$. Likewise, repeated application of the implication \eqref{eqn:implies} gives \begin{equation*} a:b::a\cdot c:b\cdot c. \end{equation*} As a special case, \begin{equation*} 1:a::b:a\cdot b. \end{equation*} Combining this with \eqref{eqn:1:a::b:ba} yields \begin{equation*} b:a\cdot b::b:b\cdot a. \end{equation*} From this we conclude \begin{equation*} a\cdot b=b\cdot a. \end{equation*} In modern symbolism and typography, such is Euclid's rigorous proof of Proposition 16 of Book \textsc{vii} of the \emph{Elements.} \section{Congruence} Let us henceforth employ the terminology and notation of Gauss, born 1777, who writes at the beginning of the \emph{Disquisitiones Arithmeticae} of 1801 \cite{Gauss-Latin}, \begin{quotation} Si numerus $a$ numerorum $b$, $c$ differantiam metitur. $b$ et $c$ \emph{secundum $a$ congrui} dicuntur, sin minus, \emph{incongrui:} ipsum $a$ \emph{modulum} appellamus. Uterque numerorum $b$, $c$ priori in casu alterius \emph{residuum,} in posteriori vero \emph{nonresiduum} vocatur\lips Omnia numeri dati $a$ residua secundum modulum $m$ sub formula $a+km$ comprehenduntur, designante $k$ numerum integrum indeterminatum\lips Numerorum congruentiam hoc signo, $\equiv$, in posterum denotabimus, modulum ubi opus erit in clausulis adiungentes, $-16\equiv9$ (mod.\ $5$), $-7\equiv15$ (mod.\ $11$). \end{quotation} In the English version of Arthur A. Clarke \cite{Gauss}, Gauss's words are rendered as follows. \begin{quotation} If a number $a$ divides the difference of the numbers $b$ and $c$, $b$ and $c$ are said to be \emph{congruent relative to} $a$; if not, $b$ and $c$ are \emph{noncongruent.} The number $a$ is called the \emph{modulus.} If the numbers $b$ and $c$ are congruent, each of them is called a \emph{residue} of the other. If they are noncongruent they are called \emph{nonresidues}\lips Given $a$, all its residues modulo $m$ are contained in the formula $a+km$ where $k$ is an arbitrary integer\lips Henceforth we shall designate congruences by the symbol $\equiv$, joining to it in parentheses the modulus when it is necessary to do so; e.\ g.\ $-7\equiv15$ (mod.\ $11$), $-16\equiv9$ (mod.\ $5$). \end{quotation} It would be more faithful to Gauss, and to his predecessors Euclid and Fermat (whom we shall consider presently), to say ``measures'' where Clarke says ``divides.'' However, we have shown that there is no mathematical difference. Where Gauss has \emph{secundum modulum,} Clarke has ``modulo.'' This is the ablative or dative case of the Latin \emph{modulus -i,} which is the diminutive of \emph{modus -i} ``measure.'' In \emph{An Adventurer's Guide to Number Theory} \cite[p.\ 116]{MR1314197}, after discussing the congruences $5\equiv 12\equiv1083\pmod7$, Richard Friedberg writes, \begin{quote} If you have studied Latin, you will understand that ``modulo $7$'' is an ablative absolute and means ``$7$ being the modulus.'' In the eighteenth century, when congruences were first studied, most mathematical articles were written in Latin. The phrase, ``modulo $7$,'' was so catchy that it still sticks. \end{quote} Friedberg is probably correct that \emph{modulo} is in the ablative case; he appears to be wrong about the reason. Again, where we say ``modulo $7$,'' Gauss says \emph{secundum modulum $7$,} ``with respect to the modulus $7$.'' \emph{Secundum} is a preposition taking the accusative case, here \emph{modulum.} The preposition derives from the adjective \emph{secundus -a -um} ``following,'' which, in the form ``second,'' serves in English as the ordinal form of the cardinal number ``two.'' When doing duty for Gauss's \emph{secundum modulum,} \emph{modulo} should probably be understood as an \emph{instrumental} ablative. The uses of the earlier Indo-European instrumental case were apparently taken up by the Latin ablative. In the present context, the modulus is the instrument---% the measuring stick---% whereby congruence is to be determined. Congruence is a geometric notion. In \emph{Number Theory and Its History} \cite[p.\ 211]{MR939614}, after defining things as Gauss does, Oystein Ore writes simply, \begin{quote} These terms, as one sees, are derived from Latin, \emph{congruent} meaning \emph{agreeing} or \emph{corresponding} while \emph{modulus} signifies \emph{little measure.} \end{quote} We can say more. Where Heath \cite{MR17:814b} translates one of Euclid's common notions as \begin{quote} Things which coincide with one another are equal to one another, \end{quote} the verb ``coincide'' might just as well be ``are congruent.'' Commandinus \cite{Euclid-Commandinus} and Heiberg \cite{Euclid-Heiberg} use the Latin source of our adjective ``congruent'' to translate Euclid's Greek, thus: \begin{quote}\centering qu\textogonekcentered e sibi ipsis congruunt, inter se sunt \textogonekcentered equalia.\\ quae inter se congruunt, aequalia sunt.\\ \gr{t`a >efarm'ozonta >ep'' >all'hla >'isa >all'hlois >est'in.} \end{quote} (Commandinus's printer uses the \textogonekcentered e or \emph{e caudata} for \emph{ae}. The printer uses also the old-fashioned long ess, when the ess is not terminal, but I have not managed to print this with \LaTeX.) \section{Divisibility} We are now allowed to use the notions of division and measurement interchangeably. We may also consider our objects of study to be not simply counting numbers, but ``signed'' counting numbers, or \emph{integers}---% of which the counting numbers are just the positive instances. Thus for example the Euclidean Algorithm allows us to find what is now called the \emph{greatest common divisor} or ``gee cee dee'' ($\gcd$) of two numbers. Moreover, the Algorithm allows us to solve the equation \begin{equation*} ax+by=\gcd(a,b), \end{equation*} where now one of $x$ and $y$ will be negative when $a$ and $b$ are positive. This result is called \textbf{B\'ezout's Lemma,} perhaps by way of impressing on students the importance of the result; such possibilities are discussed in\label{Thales} ``The Theorem of {T}hales: {A} Study of the Naming of Theorems in School Geometry Textbooks'' \cite{Thales-Theorem}, a source I used in my own study of Thales's Theorem \cite{Pierce-Thales-9}. The connection of B\'ezout to the lemma named for him does seem even more tenuous than in the case of Thales. To symbolize that an integer $a$ measures or divides an integer $b$, we may write \begin{equation*} a\divides b. \end{equation*} I do not know the origin of this notation, but Landau used it in 1927 \cite[p.\ 11]{MR0092794}, and Hardy and Wright (who also use it) say in 1938 \cite[p.\ vii]{MR568909}, \begin{quote} To Landau in particular we, in common with all serious students of the theory of numbers, owe a debt which we could hardly overstate. \end{quote} For Landau and for Hardy and Wright, unlike Gauss, the symbolism of divisibility comes before that of congruence. Hardy and Wright \cite[p.\ 49]{MR568909} observe of congruence, \begin{quote} The definition does not introduce any new idea, since `$x\equiv a\pmod m$' and `$m\divides x-a$' have the same meaning, but each notation has its advantages. \end{quote} Strictly speaking, Landau's sign of divisibility is oblique, like the solidus we use for denoting fractions. For us, $a/b$ is a rational number; for Landau, it is the assertion that $aq=b$ for some integer $q$. This assertion has the consequence that Landau expresses as $\size a/\size b$; we have to write, more confusingly, $\size a\divides\size b$. However, there are no other absolute values discussed in the present work. The fraction that for us is $a/b$ is for Landau $\frac ab$ or $a:b$. It so happens that Landau finds greatest common divisors, not with the Euclidean Algorithm, but by first observing that the least common multiple of $a$ and $b$ divides every common multiple (since otherwise the remainder would be a common multiple less than the least). The quotient of $ab$ by the least common multiple of $a$ and $b$ is shown to be the greatest common divisor. Landau and Hardy and Wright denote this by $(a,b)$, which is convenient for international use; but I shall stick with $\gcd(a,b)$. Hardy and Wright also use $\{a,b\}$, with braces, to denote the least common multiple of $a$ and $b$; but I shall use $\lcm(a,b)$. Thus \begin{equation*} \frac{ab}{\lcm(a,b)}\cdot\frac{\lcm(a,b)}b=a, \end{equation*} and similarly with $a$ and $b$ interchanged, so $ab/\lcm(a,b)$ is a common divisor of $a$ and $b$. If $d$ is a common divisor, then $ab/d$ is a common multiple, so \begin{align*} \lcm(a,b)&\;\left\vert\;\frac{ab}d,\right.& d&\;\left\vert\;\frac{ab}{\lcm(a,b)}.\right. \end{align*} Thus \begin{equation*} \gcd(a,b)=\frac{ab}{\lcm(a,b)}. \end{equation*} We shall use this and its notation once later. We shall have used braces as is customary today, to delineate sets. If $a\divides bc$ and $\gcd(a,b)=1$, then, since $ab$ is the least common multiple of $a$ and $b$, and $bc$ is \emph{some} common multiple, we have $ab\divides bc$ by what we have shown. It now follows that $a\divides c$. This is Landau's proof of what we shall call \textbf{Euclid's Lemma.} Strictly, Proposition 30 of Book \textsc{vii} of the \emph{Elements} is the case where $a$ is prime. B\'ezout's Lemma gives the neat proof of Euclid's Lemma that may be more common than Landau's. From $ax+by=1$, we obtain $acx+bcy=c$, so that, since $a\divides acx$, if also $a\divides bc$, we can conclude $a\divides c$. Useful for us at present is indeed Euclid's special case. With respect to a prime modulus $p$, if $ab\not\equiv0$, then neither of $a$ and $b$ can be congruent to $0$. This gives us \textbf{cancellation:} if $a\not\equiv0$, and $ab\equiv ac$, then $b\equiv c$. Thus the first $p-1$ multiples of $a$, starting from $a$ itself, are incongruent to one another and to $0$. Any list of numbers with this property can have length at most $p-1$. Thus if we add $1$ to the list of multiple of $a$, it must be congruent to one of these multiples. This means $a$ is \textbf{invertible} with respect to $p$. With the Euclidean Algorithm, we can actually find the inverse, since $ax+py=1$ means $ax\equiv1\pmod p$. \section{Fermat's Theorem} On Thursday, October 18, 1640, in a letter to Bernard Fr\'enicle de Bessy (1605--1675), Pierre de Fermat (1601--65) described as follows what we now know as \textbf{Fermat's Theorem} \cite[p.\ 209]{Fermat-Oeuvres-2}. \begin{quotation} Tout nombre premier mesure infailliblement une des puissances $-$ 1 de quelque progression que se soit, et l'exposant de la dite puissance est sous-multiple du nombre premier donn\'e $-$ 1; et, apr\`es qu'on a trouv\'e la premi\`ere puissance qui satisfait \`a la question, toutes celles dont les exposants sont multiples de l'exposant de la premi\`ere satisfont tout de m\`eme \`a la question. Exemple: soit la progression donn\'ee \begin{center} \begin{tabular}{*7{c}} \smaller1&\smaller2&\smaller3&\smaller4&\smaller5&\smaller6&\smaller\\ 3&9&27&81&243&729&etc. \end{tabular} \end{center} avec ses exposants en dessus. Prenez, par exemple, le nombre premier 13. Il mesure la troisi\`eme puissance $-$ 1, de laquelle 3, exposant, est sous-multiple de 12, qui est moindre de l'unit\'e que le nombre 13, et parce que l'exposant de 729, qui est 6, est multiple du premier exposant, qui est 3, il s'ensuit que 13 mesure aussi lat dite puissance 729 $-$ 1. Et cette proposition est g\'en\'eralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la d\'emonstration, si je n'appr\'ehendois d'\'etre trop long. \end{quotation} In his \emph{Source Book in Mathematics, 1200--1800} \cite[p.\ 28]{Fermat-theorem}, Struik translates Fermat as below. Instead of \emph{measures,} Struik says ``is a factor of''; instead of \emph{submultiple,} ``divisor.'' He also misdates the letter as being of October 10, 1640. \begin{quotation} Every prime number is always a factor [\emph{mesure infailliblement}] of one of the powers of any progression minus $1$, and the exponent of this power is a divisor of the prime number minus $1$. After one has found the first power that satisfies the proposition, all those powers of which the exponents are multiples of the exponent of the first power also satisfy the proposition. \emph{Example:} Let the given progression be \begin{equation*} \begin{array}{*7{r}} 1&2& 3& 4& 5& 6&\\ 3&9&27&81&243&729&\text{etc.} \end{array} \end{equation*} with its exponents written on top. Now take, for instance, the prime number $13$. It is a factor of the third power minus $1$, of which $3$ is the exponent and a divisor of $12$, which is one less than the number $13$, and because the exponent of $729$, which is $6$, is a multiple of the first exponent, which is $3$, it follows that $13$ is also a factor of this power $729$ minus $1$. And this proposition is generally true for all progressions and for all prime numbers, of which I would send you the proof if I were not afraid to be too long. \end{quotation} According to Fermat then, for every number $a$, for every prime number $p$, there is a positive exponent $\ell$ such that, with respect to the modulus $p$, \begin{equation*} a^{\ell}\equiv1; \end{equation*} moreover, if $k$ is the least such $\ell$, then $k\divides p-1$ and $a^{kx}\equiv1$ (for every multiplier $x$). Let us not fault Fermat for omitting the condition $a\not\equiv0$ and for not strictly observing that, conversely, $k$ divides every $\ell$. Usually what is called \textbf{Fermat's Theorem} is the special case that $a^{p-1}\equiv1$ when $p\ndivides a$. This is the usage of Gauss, who derives the result after proving $k\divides p-1$ as above. He then observes that can prove the basic form of Fermat's Theorem by induction. Indeed, the claim is trivially true when $a=1$. If it is true when $a=b$, then it is true when $a=b+1$, since, as a consequence of Euclid's Lemma, \begin{equation*} (b+1)^p\equiv b^p+1\pmod p. \end{equation*} Gauss attributes this proof to Euler. %\section{The Lagrange Theorem} Gauss also attributes to Euler a proof of the more general assertion of Fermat. We can summarize the proof as follows, using the terminology of Landau \cite[p.\ 42]{MR0092794}, whereby, with respect to a modulus $n$, a \textbf{complete set of residues} has any two, and therefore all three, of the following properties: \begin{compactenum}[1)] \item there are exactly $n$ members of the set, \item no two members are congruent, \item every number is congruent to one of them. \end{compactenum} If from a complete set of residues we select precisely those members that are prime to $n$, we have a \textbf{reduced set of residues.} For any prime number $p$ and any $a$ that is prime to $p$, there must be numbers $b_i$ such that the entries in the table below are incongruent from one another and compose a reduced set of residues with respect to $p$. \begin{equation*} \begin{array}{cccc} a&a^2&\cdots&a^k\\ ab_1&a^2b_1&\cdots&a^kb_1\\ ab_2&a^2b_2&\cdots&a^kb_2\\ \hdotsfor4\\ ab_n&a^2b_n&\cdots&a^kb_n \end{array} \end{equation*} In particular, the table has $k$ columns and $p-1$ entries, and therefore $k\divides p-1$. \section{Algebra} Since a reduced set of residues with respect to a given modulus is closed under both multiplication and inversion, those residues compose a finite \emph{group.} If the modulus is $n$, then the size of the group of reduced residues is the number recognized by Euler and denoted by Gauss by \begin{equation*} \euphi(n). \end{equation*} (Actually Gauss just wrote $\phi n$.) The general form of Fermat's Theorem is then a special case of the \textbf{Lagrange Theorem,} which is that the order of a finite group is divisible by the order of every subgroup. Relevant sections of Lagrange's paper \cite{Lagrange} are selected and translated in Struik's \emph{Source Book} \cite{Lagrange-Struik}; but as far as I can tell, one can infer from the paper only that the ``Lagrange Theorem'' holds when the group is the group of permutations of finitely many objects. Abstract algebra both illuminates and complicates the number theory or \emph{arithmetic} that is its origin. Number theory involves various structures, such as the groups of residues just mentioned; algebra looks at these as wholes and gives them names. As being ordered and being capable of being added and multiplied as learned in school, the integers compose a so-called \emph{ordered commutative ring,} denoted by $\Z$, supposedly for the German \textfrak{\larger Zahl.} If we are going to use the symbol $\Z$, we might as well also allow the symbol $\N$ for the positive part of $\Z$, consisting of the counting numbers. One sometimes wants a name for the \emph{non-negative} part of $\Z$, namely the positive part with zero. Some writers use $\N$ for this part, but the name $\upomega$ (omega) is already used in set theory, and so I would use that, if I had a need, which I do not in the present work. Landau and Hardy and Wright do not need even a symbol like $\Z$. The term ``ring'' for what is symbolized by $\Z$ may be unfortunate, but it seems to arise from the observation that, for example, the real numbers $a+b\surd 2$, where $a$ and $b$ are integers, also compose a ring, since the product of two such numbers ``circles back'' to being such a number as well: \begin{equation*} (a+b\surd 2)(c+d\surd 2)=ac+2bd+(ad+bc)\surd 2. \end{equation*} Given $n$ in $\N$, for the moment we let \begin{equation*} [a]_n=\{x\colon x\equiv a\}\pmod n, \end{equation*} the \textbf{congruence class} of $a$ with respect to $n$. We use this only to define \begin{equation*} \Z_n=\{[x]_n\colon x\in\Z\}, \end{equation*} the set of congruence classes with respect to $n$. Here we are only replacing each element of a complete set of residues with its congruence class. Like $\Z$ itself, $\Z_n$ is a commutative ring, though it is not ordered. The multiplicatively invertible elements---the \textbf{units}---of $\Z_n$ compose the group denoted by \begin{equation*} \units{\Z_n}. \end{equation*} Again, the size or \textbf{order} of this group is $\euphi(n)$. For every prime $p$, since $\euphi(p)=p-1$, this means both that $\Z_p$ is a \textbf{field}---% a commutative ring, like the ring of rational or real numbers, in which every nonzero element is invertible---% and that (with the help of the Lagrange Theorem) Fermat's Theorem holds. \section{Primitive roots} If $d$ and $n$ are counting numbers, $d$ dividing $n$, then the integers that have with $n$ the greatest common divisor $d$ are in one-to-one correspondence with the integers that are prime to $n/d$. The correspondence is between $dx$ and $x$, where $\gcd(x,n/d)=1$. Moreover, for every element $a$ of $\Z_n$, $\gcd(a,n)$ is well-defined, and it divides $n$. In symbols then, for all $a$ in $\Z_n$, \begin{equation}\label{eqn:iff} \gcd(a,n)=d\logicrel{if and only if} d\divides a\And\gcd\left(\frac ad,\frac nd\right)=1. \end{equation} The foregoing conclusion will serve as a lemma for a theorem that Gauss sets out \cite[\P39]{Gauss-Latin}, as we shall, in Euclid's \emph{protasis} style. I take the terminology here from David Fowler \cite[p.\ 386]{MR2000c:01004}, as naming the style's ``distinctive and useful opening feature, the enunciation or \emph{protasis.}'' In a proposition of Euclid, first comes an enunciation, and only then comes the demonstration or proof of what has been enunciated. Students readily adopt this style, first writing what they want to prove, then writing down more things, which they hope will be considered as a proof. It is not always clear that the students understand the logical relations involved. Euclid avoids confusing his readers by following a consistent pattern. Each of his propositions has up to six parts, always in the same order. In his commentary on Euclid, Proclus names the parts of a proposition as enunciation (\gr{pr'otasis}), exposition, specification, construction, demonstration, and conclusion \cite[p.\ 159 (203)]{MR1200456}. Neither Proclus nor Euclid has a name for what we call the proposition as a whole. Proclus says, in Morrow's translation \cite[p.\ 63 (77)]{MR1200456}, \begin{quote} Again the propositions that follow from the first principles he divides into problems and theorems; \end{quote} but as in the King James Bible, the words ``propositions that follow'' could be italicized, as having no explicit counterpart in the Greek, which, for the passage just quoted, is \cite[p.\ 77]{1873procli} \begin{quote} \gr{P'alin d'' a>~u t`a >ap`o t~wn >arq~wn e>is probl'hmata diaire~itai ka`i jewr'hmata.} \end{quote} Pappus makes the etymology clear \cite[pp.\ 566 f.]{MR13:419b}: in a problem (\gr{pr'oblhma}) it is proposed (\gr{prob'alleta'i}) to do something; in a theorem (\gr{je'wrhma}), the implications of hypotheses are contemplated (\gr{jewre~itai}). Euclid signals the distinction between a problem and a theorem by how he ends it, using respectively the words that we translate into Latin and abbreviate as \textsc{q.e.f.}\ (``which was to be done'') and \textsc{q.e.d.}\ (``which was to be proved''). A web edition of Euclid's Greek text labels each proposition as a \gr{pr'otasis} \cite{Euclid-homepage}. This conforms to the modern practice of treating the enunciation as a \emph{metonym} for the proposition as a whole. Here the terminology is from Reviel Netz, who argues that for Euclid the \emph{diagram} is the metonym of the proposition \cite[p.\ 38]{MR1683176}. The handy little book called \emph{The Bones} \cite{bones} does not choose sides, but supplies both the enunciation and the diagram (and nothing else) for each of Euclid's propositions. For the proposition below, Gauss simply italicizes his protasis: \begin{quote}\itshape Si $a$, $a'$, $a''$, etc.\ sunt omnes divisores ipsius $A$ (unitate et ipso $A$ non exclusis), erit \begin{equation*} \phi a+\phi a'+\phi a''+\text{etc.}= A. \end{equation*} \end{quote} We give the protasis a bold label. \begin{theorem}[Gauss] The sum of the values of $\euphi(d)$, as $d$ ranges over the positive divisors of $n$, is just $n$ itself; in symbols, \begin{equation}\label{eqn:Gauss} \sum_{d\divides n}\euphi(d)=n. \end{equation} \end{theorem} \begin{proof} Suppose $d\divides n$. By \eqref{eqn:iff}, the two sets \begin{align*} &\{x\in\Z_n\colon\gcd(x,n)=d\},&&\{y\in\Z_{n/d}\colon\gcd(y,n/d)=1\} \end{align*} have the same size. The size of the latter set being $\euphi(n/d)$, we can conclude \begin{equation*} n=\sum_{d\divides n}\euphi\left(\frac nd\right). \end{equation*} This yields \eqref{eqn:Gauss}, by symmetry. \end{proof} The \textbf{order} of an element $a$ of $\units{\Z_n}$ is the least positive exponent $\ell$ such that $a^{\ell}=1$. In symbols, \begin{equation*} \ord na=\min\{x\in\N\colon a^x=1\}. \end{equation*} If we think of $a$ as an integer, rather than a congruence class, we should perhaps write something like \begin{equation*} \ord na=\min\{x\in\N\colon a^x\equiv1\}\pmod n. \end{equation*} If it exists, a \textbf{primitive root} of $n$ is an element of $\units{\Z_n}$ having order $\euphi(n)$. Euler gave a proof of the following, but there was a gap, which, according to Burton \cite[p.\ 162]{Burton}, Legendre filled. Gauss mentions the gap, but not Legendre. \begin{theorem} Every prime number has a primitive root. \end{theorem} \begin{proof} Let $p$ be a prime number. We shall show that the number of its primitive roots is $\euphi(p-1)$, which is positive. Since $\euphi(p)=p-1$, the order of every element of $\units{\Z_p}$ measures this. If $d\divides p-1$, let us denote by \begin{equation*} \uppsi_p(d) \end{equation*} the number of elements of $\units{\Z_p}$ having order $d$. Since every element of $\units{\Z_p}$ has some such order $d$, we have \begin{equation*} p-1=\sum_{d\divides p-1}\uppsi_p(d). \end{equation*} By the previous theorem, we shall be done when we show $\uppsi_p(d)\leq\euphi(d)$. Suppose $\uppsi_p(d)>0$, so that some $a$ in $\units{\Z_p}$ has order $d$. The $d$ elements of the set $\{a^t\colon t\in\Z_d\}$ are solutions of the congruence \begin{equation*} x^d\equiv1\pmod p. \end{equation*} They must be the only solutions, since the congruence can have at most $d$ solutions (since $\Z_p$ is a field). Moreover, with respect to $p-1$, \begin{align*} \ord p{a^k} &=\min\{x\in\N\colon a^{kx}\equiv1\}\\ &=\frac1k\min\{y\in\N\colon k\divides y\And a^y\equiv1\}\\ &=\frac1k\min\{y\in\N\colon k\divides y\And d\divides y\}\\ &=\frac{\lcm(k,d)}k, \end{align*} and this is $d/\gcd(k,d)$, by what we showed earlier. Thus, if it is positive, $\uppsi_p(d)$ must be the size of the set \begin{equation*} \{x\in\Z_d\colon\gcd(x,d)=1\}, \end{equation*} and this size is by definition $\euphi(d)$. \end{proof} The foregoing is the \emph{first} of Gauss's two proofs. It is the one that Hardy and Wright give \cite[pp.\ 85 f.]{MR568909}; but they give it, unlike Gauss, \emph{after} proving Gauss's Law of Quadratic Reciprocity. Like other writers, they follow Gauss in using the notation $\psi$ where I have $\uppsi_p$. It seems to me desirable to use the subscript $p$, so that the ultimate independence of $\uppsi_p(d)$ from $p$ (as long as $d\divides p-1$) may be all the more remarkable. I also make the subtle distinction of using an upright letter $\uppsi$ for something defined once for all; an italic letter like $\psi$ may have different meanings in different settings, even though the meaning may be considered constant in a particular setting. \section{Two more proofs} %\begin{sloppypar} Landau gives Gauss's second proof that primes have primitive roots, but he gives it, like Hardy and Wright, only after Quadratic Reciprocity. It uses the Fundamental Theorem of Arithmetic, that every prime number has a unique prime factorization. Gauss seems to have been the first person to state this explicitly \cite[p.\ 10]{MR568909}. %\end{sloppypar} Briefly, suppose $p-1$ has the prime factorization $\prod_qq^{d(q)}$. For each prime $q$ in the product, since the congruence \begin{equation*} x^{(p-1)/q}\equiv1\pmod p \end{equation*} has at most $(p-1)/q$ solutions, it has a non-solution, $a_q$, from $\units{\Z_p}$. Then $q^{d(q)}$ is the order of the power \begin{equation*} a_q{}^{(p-1)/q^{d(q)}}, \end{equation*} and the product $\prod_qa_q{}^{(p-1)/q^{d(q)}}$ of all of these powers has order $p-1$. For a third proof that every prime number has a primitive root, noting as we have that $\Z_p$ is a field, we can just prove generally that the group of units of every finite field $K$ is \emph{cyclic.} Again briefly, if $a$ and $b$ in $\units K$ have orders $k$ and $m$, then $ab$ must have order $km$, if $k$ and $m$ are prime to one another. As a result, if $\gcd(k,m)=d$, then $a^{k/d}b$ has order $\lcm(k,m)$. If $a$ already has maximal order, then $k\divides m$. In this case, every element of $\units K$ is a root of the polynomial $x^m-1$; in a field, this can have no more than $m$ roots; therefore $m$ is less by $1$ than the size of $K$. \section{Practicalities} The number 997 is prime, because (1) it is less than 1024, which is the square of 32, and (2) it is indivisible by the primes less than 32, namely 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. Now we know that 997 has a primitive root, and Gauss's second proof suggests a procedure for finding one. Alternatively, if $a$ is a candidate, since 996 has the prime factorization $2^2\cdot3\cdot83$, it is enough to check that none of the powers \begin{align*} &a^{2^2\cdot3},&&a^{2^2\cdot83},&&a^{3\cdot83} \end{align*} is congruent to unity. One can compute these by hand by taking successive squares and using for example \begin{equation*} 83=64+16+2+1=2^6+2^4+2^2+2^0. \end{equation*} In fact a table of primes and their primitive roots in Burton \cite[p.\ 393]{Burton} gives 7 as the least primitive root of 997. 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