“Voici ce que j’ai trouvé ” Sophie Germain’s grand plan to prove Fermat’s Last Theore

“V oici ce que j’ai trouvé:” Sophie Germain’s

grand plan to prove Fermat’s Last Theorem y

Reinhard Laubenbacher

Virginia Bioinformatics Institute

Virginia Polytechnic Institute and State University

Blacksburg,V A24061,USA

David Pengelley z

Mathematical Sciences

New Mexico State University

Las Cruces,NM88003,USA

Copyright c 2007Reinhard Laubenbacher&David Pengelley

July4,2007

Abstract

A study of Sophie Germain’s extensive manuscripts on Fermat’s

Last Theorem calls for a reassessment of her work in number theory.

There is much in these manuscripts beyond the single theorem for

Case1for which she is known from a published footnote by Legendre.

Germain had a fully-?edged,highly developed,sophisticated plan of

attack on Fermat’s Last Theorem.The supporting algorithms she

invented for this plan are based on theoretical concepts,ideas and “Here is what I have found:”

y We owe heartfelt thanks to many people who have helped us tremendously with this project over a long…fteen years:Hélène Barcelo,Louis Bucciarelli,Keith Dennis,Mai Gehrke,Tiziana Giorgi,Catherine Goldstein,Maria Christina Mariani,Pat Pen…eld,Do-nato Pineider,and Ed Sandifer,along with Marta Gori of the Biblioteca Moreniana, as well as the Bibliothèque Nationale,New York Public Library,Nieders?chsische Staats-und Universit?tsbibliothek G?ttingen,and the Interlibrary Loan sta¤of New Mexico State University.

z Dedicated to the memory of my parents,Daphne and Ted Pengelley,for inspiring my interest in history,and to Pat Pen…eld,for her talented,dedicated,and invaluable editorial help,love and enthusiasm,and support for this project.

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results discovered independently only much later by others,and her

methods are quite di¤erent from any of Legendre’s.In addition to her

program for proving Fermat’s Last Theorem in its entirety,Germain

also made major e¤orts at proofs for particular families of exponents.

The isolation Germain worked in,due in substantial part to her di￠cult

position as a woman,was perhaps su￠cient that much of this extensive

and impressive work may never have been studied and understood by

anyone.

Uneétude approfondie des manuscrits de Sophie Germain sur le dernier théorème de Fermat,révèle que l’on doit réévaluer ses travaux

en théorie des nombres.En e¤et,on trouve dans ses manuscrits beau-

coup plus que le simple théorème du premier cas que Legendre lui

avait attribuédans une note au bas d’une page et pour lequel elle est

reconnue.Mme Germain avait un plan trèsélaboréet sophistiquépour

prouver entièrement ce dernier théorème de Fermat.Les algorithmes

qu’elle a inventés sont basés sur des concepts théoriques qui ne furent

indépendamment découverts que beaucoup plus tard.Ses méthodes

sontégalement assez di¤érentes de celles de Legendre.En plus,Mme

Germain avait fait de remarquables progrès dans sa recherche concer-

nant certaines familles d’exposants.L’isolement dans lequel Sophie

Germain se trouvait,en grande partie d?au fait qu’elleétait une

femme,fut peut-être su￠sant,que ses impressionnants travaux au-

raient pu passer complètement inaper?us et demeurer incompris.

Das Studium von Sophie Germains extensiven Manuskriptenüber den Fermat Satz legt eine Neuauslegung ihrer zahlentheoretischen Ar-

beiten nahe.Diese Manuskripte enthalten viel mehr als nur das einzige

Theoremüber Fall I für das sie aufgrund einer ver?¤entlichten Fuss-

note von Legendre bekannt ist.Germain hatte einen umfassenden,

ausgereiften und tiefgehenden Angri¤splan für das Fermat Problem.

Die zugrundeliegenden Algorithmen die sie dafür erfand basieren auf

theoretischen Konzepten,Ideen und Resultaten die erst viel sp?ter von

anderen unabh?ngig wiederentdeckt wurden,und ihre Methoden un-

terscheiden sich deutlich von denen Legendres.über ihr Programm das

Fermatsche Problem komplett zu l?sen hinaus hat Germain auch grosse

Anstrengungen gemacht Beweise für einzelne Familien von Exponen-

ten zu…nden.Die Isolation in der Germain arbeitete war vielleicht

genug dass vieles dieses ausgreifenden und beeindruckenden Werkes

der Nachwelt verloren h?tte gehen koennen.

Contents

1Introduction4

1.1Gauss and Germain on number theory (6)

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1.2Sophie Germain’s explication to Gauss of her grand plan (8)

1.3Our manuscript sources (10)

1.4The major divisions of Germain’s work (12)

1.5Reevaluation (13)

2Sophie Germain’s grand plan13

2.1Germain’s plan for proving Fermat’s Last Theorem (14)

2.2Did Germain ever know her grand plan cannot succeed? (18)

2.3Comparing Germain’s grand plan with Legendre,Dickson,

and recent results on Case1 (23)

2.4Comparing Manuscripts A and D:Polishing for the prize com-

petition? (26)

3Large size of solutions27

3.1Germain’s approach to large size solutions (28)

3.2Comparing Germain on Condition p-N-p and Large Size with

Legendre,Wendt,Dickson,Vandiver (36)

4Fermat’s Last Theorem for exponents of form2(8n 3)39

4.1Case1and Sophie Germain’s Theorem (40)

4.2Case2for p dividing z (42)

4.3Case2for p dividing x or y (43)

4.4Manuscript B as source for Legendre? (43)

5Fermat’s Last Theorem for even exponents44 6Germain’s approaches to Fermat’s Last Theorem:précis and connections45

6.1The grand plan to prove Fermat’s Last Theorem (45)

6.2Large size of solutions and Sophie Germain’s Theorem (45)

6.3Exponents2(8n 3)and Sophie Germain’s Theorem (46)

6.4Even exponents (46)

7Reevaluation of Germain’s work in number theory47

7.1Germain as strategist:theories and techniques (47)

7.2Interpreting the errors in Germain’s manuscripts (48)

7.3Review by others versus isolation (49)

8Conclusion52

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1Introduction

Sophie Germain was the…rst woman known for important original research in mathematics.1While Germain is perhaps best known for her work in mathematical physics,her number theoretic research on Fermat’s Last The-orem has been considered by many to be her best mathematics.We will make a substantial reevaluation of her work on the Fermat problem,based on translation and detailed mathematical interpretation of numerous doc-uments in her own hand,heretofore perhaps never seriously analyzed,and will argue that her accomplishments are much broader,deeper,and more signi…cant than has ever been realized.

On the twelfth of May,1819,Sophie Germain penned a letter from her Parisian home to Carl Friedrich Gauss in G?ttingen[Ge1].Most of this lengthy letter describes in some detail her work on substantiating Pierre de Fermat’s claim that the equation z p=x p+y p has no solutions in positive natural numbers for exponents p>2.The challenge of proving this famous assertion of Fermat has had a tumultuous history,culminating in Andrew Wiles’success at the end of the twentieth century[Ri].

Shortly we shall see what astonishing words Germain wrote to Gauss in her letter of1819,but…rst let us brie?y recap,for both context and contrast, exactly what she has been known for from the number theory literature.

Once Fermat’s claim had been proven by Euler for exponent4in the eighteenth century,it could be fully con…rmed by substantiating it just for odd prime exponents.But when Germain began working on the problem at the turn of the nineteenth century,the only prime exponent that had a proof was3[Ed,Ri].In1823Adrien-Marie Legendre wrote a treatise on Fermat’s Last Theorem,ending with his own ad hoc proof for exponent 5.What interests us,though,is the…rst part of Legendre’s treatise,since Germain’s work on the Fermat problem has long been understood to be entirely described by a single footnote there[Di,Ed,Le,Ri].Here Legendre presents a general analysis of the Fermat equation,whose main theoretical highlight is a theorem encompassing all odd prime exponents.In modern terminology:

Sophie Germain’s Theorem.For an odd prime exponent p,if there exists an auxiliary prime such that there are no two nonzero consecutive p-th powers modulo ,nor is p itself a p-th power modulo ,then in any solution 1A good biography of Germain,with concentration on her work in elasticity theory, discussion of her personal and professional life,and references to the historical literature about her,is the book by Louis Bucciarelli and Nancy Dworsky[BD].

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to the Fermat equation z p=x p+y p,one of x,y,or z must be divisible by p2.

Legendre supplies a table verifying the hypotheses of the theorem for p<100by brute force display of all the p-th power residues modulo a single auxiliary prime chosen for each value of p.Legendre then credits Sophie Germain with both the theorem,which is the…rst general result about ar-bitrary exponents for Fermat’s Last Theorem,and its successful application for p<100.One assumes from Legendre that Germain developed the brute force table of residues as her means of veri…cation and application of her the-orem.Legendre continues on to develop more theoretical means of verifying the hypotheses of Sophie Germain’s Theorem,and he also pushes the analy-sis further to demonstrate that any solutions to certain Fermat equations would have to be extremely large.

For almost two centuries,it has been assumed that this theorem and its brute force tabular application to exponents less than100constitute Germain’s entire work in this area,the basis of her reputation[Ed,Ri]. However,we will…nd that this presumption is dramatically o¤the mark as we study Germain’s manuscripts.It is not easy to decipher Germain’s handwriting,translate,…ll in gaps,and understand both the grammar and the mathematics of the extant archive material in Germain’s hand.But the reward is a wealth of new material,a vast expansion over the very limited information known just from Legendre’s footnote.We will explore the much enlarged scope and extent of Germain’s work that is revealed, and its ambitiousness and importance.Together these will prompt a major reevaluation,and recommend a substantial elevation of her reputation.

Before going directly to Germain’s own writing,we note that even the historical record based solely on Legendre’s footnote has been unjustly por-trayed.Even the limited results that Legendre clearly attributed to Ger-main have been badly understated and misattributed in much of the vast secondary literature.Some writers state only weaker forms of Sophie Ger-main’s Theorem,such as merely for p=5,or only for auxiliary primes of the form2p+1(known as“Germain primes”,which happen always to satisfy the two required hypotheses).Others only conclude divisibility by the…rst power of p,and some writers have even attributed the fuller p2-divisibility, or the determination of qualifying auxiliaries for p<100,to Legendre rather than to Germain.A few have even confused the results Legendre credited to Germain with a completely di¤erent claim she had made in her…rst letter to Gauss,in1804[St].Fortunately a few books have correctly stated Legen-dre’s attribution to Germain[Di,Ed,Ri].We will not elaborate in detail on

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the huge related mathematical literature except for speci…c relevant com-parisons of mathematical content with Germain’s own work.Ribenboim’s most recent book[Ri]gives a good overall history of related developments, including windows into the large intervening mathematical literature.

In spite of the failures of much of the literature to report accurately the credit Legendre gave her,Sophie Germain’s Theorem can clearly be used, by producing a valid auxiliary,to eliminate the existence of solutions to the Fermat equation involving numbers not divisible by the exponent p.This elimination is today called“Case1”of Fermat’s Last Theorem.Work on Case1has continued to the present,and major results,including for instance its recent establishment for in…nitely many prime exponents p[AH,Fo],have been proven by building on the very theorem that Germain introduced. 1.1Gauss and Germain on number theory

Let us compare the meager published historical record,responsible for her reputation,with Germain’s own words to Gauss.Her1819letter was written after an eleven year hiatus in their correspondence,so she had much to catch up on.The letter describes the broad scope of Germain’s many years of work,in addition to much detail on her program for proving Fermat’s Last Theorem:

11111111

[...]Although I have worked for some time on the theory of vibrating surfaces [...],I have never ceased thinking about the theory of numbers.I will give you a sense of my absorption with this area of research by admitting to you that even without any hope of success,I still prefer it to other work which might interest me while I think about it,and which is sure to yield results.

Long before our Academy proposed a prize for a proof of the impossibility of the Fermat equation,this type of challenge,which was brought to modern theories by a geometer who was deprived of the resources we possess today, tormented me often.I glimpsed vaguely a connection between the theory of residues and the famous equation;I believe I spoke to you of this idea a long time ago,because it struck me as soon as I read your book.2

11111111

2“Quoique j’ai travaillépendant quelque tem[p]s a làthéorie des surfaces vibrantes [:::],je n’ai jamais cesséde penser a la théorie des nombres.Je vous donnerai une ideéde ma préoccupation pour ce genre de recherches en vous avouant que même sans aucune esperance de succès je la prefere a un travail qui me donnerais necessairement en resultat et qui pourtant m’interresse:::quand j’y pense.

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Clearly number theory held a very special fascination for Germain through-out much of her https://www.sodocs.net/doc/0d16220326.html,rgely self-taught,due to her exclusion as a woman from higher education and normal subsequent academic life,she…rst stud-ied Legendre’s Théorie des Nombres,published in1789,and then devoured Gauss’Disquisitiones Arithmeticae when it appeared in1801.Gauss’work was a complete departure from everything that came before,and established number theory as a mathematical subject in its own right,with its own body of methods and techniques,such as the theory of congruences.Germain ini-tiated a correspondence with Gauss,initially under the male pseudonym LeBlanc,which continued for a number of years and gave tremendous impe-tus to her work.In this early exchange of letters lasting from1804to1808, she sent Gauss some of her work on Fermat’s Last Theorem stemming from inspiration she had received from his Disquisitiones.Excerpts can be found in Chapter3of[BD]and in[St].

Gauss was greatly impressed by Germain’s work,and was even stimu-lated thereby in some of his own,as evidenced by his remarks in a number of letters to his colleague Wilhelm Olbers.On September3,1805Gauss wrote[Sc,p.268]:“Through various circumstances—partly through sev-eral letters from LeBlanc in Paris,who has studied my Disq.Arith.with a true passion,has completely mastered them,and has sent me occasional very respectable communications about them,[:::]I have been tempted into resuming my beloved arithmetic investigations.”After LeBlanc’s true iden-tity was revealed to him,he writes again to Olbers,on March24,1807[Sc, p.331]:“Recently my Disq.Arith.caused me a great surprise.Have I not written to you several times already about a correspondent LeBlanc from Paris,who has given me evidence that he has mastered completely all in-vestigations in this work?This LeBlanc has recently revealed himself to me more closely.That LeBlanc is only a…ctitious name of a young lady Sophie Germain surely amazes you as much as it does me.”

Gauss’letter of July21of the same year shows that Germain was a valued member of his circle of correspondents[Sc,pp.376–377]:“Upon my return I have found here several letters from Paris,by Bouvard,Lagrange, and Sophie Germain.[:::]Lagrange still shows much interest in astronomy and higher arithmetic;the two sample theorems(for which prime numbers “Longtems[sic]avant que notre academie ais proposépour sujet de prix la démon-stration de l’impossibilitéde l’équation de Fermat ces espece de dé…—portéaux théories modernes par un géometre—qui fus privédes resources que nous possedons aujourd’hui me tourmentois souvent.Y’entrevoyais vaguement une liaison entre la théorie des residues et la fameuseéquation,je crois même vous avoir parléanciennement de cette ideécar elle m’a frappéaussit?t que j’ai connu votre livre.”

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is two a cubic or biquadratic residue),which I also told you about some time ago,he considers‘that which is most beautiful and di￠cult to prove.’But Sophie Germain has sent me the proofs for them;I have not yet been able to look through them,but I believe they are good;at least she has approached the matter from the right point of view,only they are a little more long-winded than will be necessary.”

The two theorems on power residues were part of a letter Gauss wrote to Germain on April30,1807[Ga1,vol.10,pp.70–74].Together with these theorems he also included,again without proof,another result now known as Gauss’Lemma,from which he says one can derive special cases of the Quadratic Reciprocity Theorem.In a May12,1807letter to Olbers,Gauss says“Recently I replied to a letter of hers and shared some Arithmetic with her,and this led me to undertake an inquiry again;only two days later I made a very pleasant discovery.It is a new,very neat,and short proof of the fundamental theorem of art.131.”[Ga1,vol.10,p.566]The proof Gauss is referring to,based on the above lemma in his letter to Germain,is now commonly called his“third”proof of the Quadratic Reciprocity Theorem, and was published in1808[Ga2],where he says he has…nally found“the simplest and most natural way to its proof”(see also[LP1,LP2]).

1.2Sophie Germain’s explication to Gauss of her grand plan Germain continues the letter of1819to Gauss by explaining her major e¤ort to prove Fermat’s Last Theorem:

11111111

Here is what I have found:[...]

The order in which the residues(powers equal to the exponent)are distrib-uted in the sequence of natural numbers determines the necessary divisors which belong to the numbers among which one establishes not only the equation of Fermat,but also many other analogous equations.

Let us take for example the very equation of Fermat,which is the simplest of those we consider here.Therefore we have z p=x p+y p,p a prime number.

I claim that if this equation is possible,then every prime number of the form 2Np+1(N being any integer),for which there are no two consecutive p-th power residues in the sequence of natural numbers,necessarily divides one of the numbers x,y,and z.

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This is clear,since[if not]the equation z p=x p+y p yields the congruence 1 r sp r tp in which r represents a primitive root and s and t are integers.3 [...]

It follows that if there were in…nitely many such numbers,the equation would be impossible.

I have never been able to arrive at the in…nity,although I have pushed back the limits quite far by a method of trials too long to describe here.I still dare not assert that for each value of p there is no limit beyond which all numbers of the form2Np+1have two consecutive p-th power residues in the sequence of natural numbers.This is the case which concerns the equation of Fermat.

You can easily imagine,Monsieur,that I have been able to succeed at proving that this equation is not possible except with numbers whose size frightens the imagination;because it is also subject to many other conditions which I do not have the time to list because of the details necessary for establishing??????. But all that is still not enough;it takes the in…nite and not merely the very large.4

11111111

3Germain is considering congruence modulo the auxiliary prime =2Np+1.She is observing that if none of x;y;z were divisible by ,then division of the Fermat equation by x p or y p would produce two nonzero consecutive p-th power residues.Her claim follows.

4“Voici ce que j’ai trouvé:

“L’ordre dans lequel les residus(puissances egales a l’exposant)se trouvent placés dans la serie des nombres naturels détermine les diviseurs necessaires qui appartiennens aux nombres entre lequels onétablis non seulement l’équation de Fermat mais encore beaucoup d’autreséquations analogues a celle là.

“Prenons pour example l’équation même de Fermat qui est la plus simple de toutes celles dont il s’agit ici.Soit donc,pétant un nombre premier,z p=x p+y p.Je dis que si cetteéquation est possible,tous[sic]nombre premier de la forme2Np+1(Nétant un entier quelconque)pour lequel il n’y aura pas deux résidus p ièm e puissance placés de suite dans la serie des nombres naturels divisera nécessairement l’un des nombres x y et z.“Cela estévident,car l’équation z p=x p+y p donne la congruence1 r sp r tp dans laquelle r represente une racine primitive et s et t des entiers.

“:::Il suis delàque s’il y avois un nombre in…ni de tels nombres l’équation serois impossible.

“Je n’ai jamais p?arriver a l’in…ni quoique j’ai reculébien loin les limites par une methode de tatonnement trop longue pour qu’il me sois possible de l’exposer ici.Je n’oserois même pas a￠rmer que pour chaque valeur de p il n’existe pas une limite audela delaquelle tous les nombres de la forme2Np+1auroient deux résidus p ièm es placés de suite dans la serie des nombres naturels.C’est le cas qui interesse l’équation de Fermat.“Vous concevrez aisement,Monsieur,que j’ai d?parvenir a prouver que cetteéquation ne serois possible qu’en nombres dont la grandeur e¤raye l’imagination;Car elle est encore assujettée a bien d’autres conditions que je n’ai pas le tems[sic]d’énumerer a cause des details necessaire pour enétablir?lu??reassité?.Mais tout cela n’est encore rien,il faut l’in…ni et non pas le très grand.”

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Several things are remarkable here.Most surprisingly,Germain does not mention to Gauss anything even hinting at the only result she is actually known for in the literature,what we call Sophie Germain’s Theorem.Why not?Where is it?Instead,Germain explains a plan,simple in its conception, for proving Fermat’s Last Theorem outright.It requires that,for a given prime exponent p,one establish in…nitely many auxiliary primes each satis-fying a non-consecutivity condition on its p-th power residues(note that this condition is the very same as one of the two hypotheses required in Sophie Germain’s Theorem for proving Case1,but there one only requires a single auxiliary prime,not in…nitely many).She writes that she has worked long and hard at this plan by developing a method for verifying the condition, made great progress,but has not been able to bring it fully to fruition(even for a single p)by verifying the condition for in…nitely many auxiliary primes. She also writes that she has proven that any solutions to a Fermat equation would have to frighten the imagination with their size.And she explains in broad outline all her work on the problem.Clearly we should now be very curious about her work in these two directions,perhaps completely distinct from the theorem for which Legendre cites her.

1.3Our manuscript sources

Fortunately,the Germain biography[BD],which led us to her letter to Gauss,also tells us that many of Germain’s manuscripts lie in the archives of the Bibliothèque Nationale in Paris.Many others are also held in the Bib-lioteca Moreniana,in Firenze(Florence),Italy[Ce,Ce1].The story of how Germain’s manuscripts ended up in these two collections is an extraordi-nary and fascinating one,a consequence of the amazing career of Guglielmo (Guillaume)Libri,mathematician,historian,bibliophile,thief,and friend of Sophie Germain[Ce,RM].In particular,it appears that many of Germain’s manuscripts in the Bibliothèque Nationale were probably among those con-…scated by the police from Libri’s apartment at the Sorbonne when he?ed to London in1848to escape the charge of thefts from French public libraries [Ce,p.146].The Germain manuscripts in the Biblioteca Moreniana were among those shipped with Libri’s still remaining vast collection of books and manuscripts before he set out to return from London to Florence in1868. The Germain materials are among those fortunate to have survived intact despite a tragic string of events following Libri’s death in1869[Ce,Ce1]5.

5See also[Ce2,Ce3,Ce4]for the fascinating story of Abel manuscripts discovered in the same Libri collections.

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How Libri obtained Germain’s manuscripts remains unknown,but it would be entirely in character for him to have managed this,since by hook

or by crook he built a gargantuan private library of important books,man-uscripts,and letters[Ce].6We should probably thank Libri,the e¤orts of many others after his death,and much good fortune,for saving Germain’s amazing manuscripts.Otherwise they might well simply have drifted into oblivion.

There are hundreds of sheets of Germain’s handwritten papers in the Bibliothèque Nationale,many of them number theory.They are almost

all undated,relatively unorganized,almost all unnumbered except by the archive.And they range all the way from scratch paper to some beautifully polished…nished pieces,in handwriting that is sometimes extremely di￠-

cult to decipher.It appears that their mathematical content has received little attention in the nearly two centuries since Germain wrote them.We cannot possibly provide a de…nitive evaluation here of this treasure trove

of Germain’s manuscripts in the Bibliothèque Nationale,as well as those

in the Biblioteca Moreniana.Rather,we will focus our attention in these manuscripts on the major claims she made in her1819letter to Gauss,their potential relationship to Sophie Germain’s Theorem,and her other work on Fermat’s Last Theorem.

We will explain some of Germain’s most important mathematical ap-proaches to Fermat’s Last Theorem,provide a sense for the results she successfully obtained,compare them with the impression of her work left

by Legendre’s treatise,and in particular discuss possible overlap between Germain’s work and Legendre’s.We will also…nd connections between Ger-main’s work on Fermat’s Last Theorem and that of mathematicians of the later nineteenth and twentieth centuries.Finally,we will discuss claims in Germain’s manuscripts to have actually fully proven Fermat’s Last Theorem

for certain exponents.

The assessment presented here is based principally on study of her two undated manuscripts entitled Remarques sur l’impossibilitéde satisfaire en nombres entiers a l’équation x p+y p=z p[Ge2,pp.198(right)–208(left)] (hereafter called Manuscript A,20sheets numbered in her hand,but at-tached two to one to the archive numbering),and Démonstration de l’impossibilitéde satisfaire en nombres entiersàl’équation z2(8n 3)=y2(8n 3)+x2(8n 3) 6After his expulsion from Tuscany for his role in the plot to persuade the Grand-Duke

to promulgate a constitution,Libri traveled for many months,not reaching Paris until

fully six months after Germain died,making it all the more extraordinary that it seems

he ended up with almost all her papers.[Ce,p.142f]

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[Ge2,pp.92(right)–94(left)](Manuscript B,4sheets)7,along with a pol-ished set of three pages[Ge3,p.348(right)–349(right)](Manuscript C) stating and claiming a proof of Fermat’s Last Theorem for all even expo-nents.These three manuscripts are found in the Bibliothèque Nationale. We will also compare Manuscript A with another very similar manuscript of the same title,held in the Biblioteca Moreniana(Manuscript D,25pages, the19th blank;see our discussion)[Ge5,cass.11,ins.266][Ce,p.234]. Together these appear to be her primary pieces of polished work in these archives on Fermat’s Last Theorem.Nevertheless,our assessment is based on only part of her approximately150–200pages of number theory manu-scripts in the Bibliothèque,and other researchers may ultimately have more success than we at deciphering,understanding,and interpreting them.Also, there are numerous additional Germain papers in the Biblioteca Moreniana that may yield further insight[Ce,Ce1].Finally,even as our analysis and evaluation answers many questions,it will also raise numerous new ones,so there is fertile ground for much more study of her manuscripts by others. In particular,questions of the chronology of much of her work,and of her interaction with others,still contain enticing perplexities.

1.4The major divisions of Germain’s work

In section2we will elucidate from Manuscripts A and D the methods Ger-main developed in her“grand plan”for proving Fermat’s Last Theorem outright,the progress she made,and its di￠culties.We will compare Ger-main’s methods with her explanation to Gauss and Legendre’s work.More-over,the non-consecutivity condition on p-th power residues,which is key to both Germain’s grand plan and to utilizing her theorem to prove Case1,has been pursued by later mathematicians all the way to the present day,and we will compare her approach to later ones.We will also explore whether Germain at some point realized that her grand plan could not be carried through,using the published historical record and a single relevant letter from Germain to Legendre.

In section3we will explore Germain’s e¤ort at proving and applying a theorem which we shall call“Large size of solutions”,whose intent is to convince that any solutions which might exist to a Fermat equation would have to be astronomically large,the second point she mentioned to Gauss. Her e¤ort is challenging to evaluate,since her proof as given in the pri-mary manuscript is?awed,but she later recognized this and attempted to 7Part of this manuscript,essentially the content of Sophie Germain’s Theorem,was translated and discussed in[LP].

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compensate.Moreover Legendre published similar results and applications, which we will contrast with Germain’s.We will discover that the theorem which has been known in the literature as Sophie Germain’s Theorem is simply minor fallout from her“Large size of solutions”analysis,and we compare some of the methods used by later workers to apply her theorem with her own methods.

Germain’s mathematical aims,namely her grand plan,large size of solu-tions,and p2-divisibility(which includes Case1),are all intertwined in her manuscripts,largely because the hypotheses needing veri…cation overlap. We have separated our exposition of them,however,in order to facilitate direct comparison with Legendre’s treatise,which had a di¤erent focus but much apparent overlap with Germain’s,and to enable easier comparison with the later work of others.

In section4we will analyze Manuscript B,which claims proof of Fermat’s Last Theorem for a large family of exponents,by building on an essentially self-contained statement of Sophie Germain’s Theorem.And in section5 we consider a very di¤erent approach(Manuscript C)claiming to prove Fermat’s Last Theorem for all even exponents,based on the impossibility of another Diophantine equation.

1.5Reevaluation

Our paper will end with an assessment of Germain’s full-?edged attack on Fermat’s Last Theorem,her analysis leading to claims of astronomical size for any possible solutions to the Fermat equation,the fact that Sophie Ger-main’s Theorem is in the end a small piece of something much more ambi-tious,our assessment of how independent her work actually was from her mentor Legendre’s,of the methods she invented for verifying various condi-tions,and the paths unknowingly taken in her footsteps by later researchers. We propose that a substantial reevaluation is in order.The results Sophie Germain obtained and the methods she developed place her at the forefront of number-theoretic research in the early nineteenth century.

2Sophie Germain’s grand plan

In her letter to Gauss[Ge1]in1819,Germain summarized her plan for prov-ing Fermat’s Last Theorem.Our aim is to show its promise,thoroughness and sophistication.

Manuscript A contains,among other things,the details of this program for producing,for each odd prime exponent p,an in…nite sequence of quali-

13

fying auxiliary primes,which,as she explained to Gauss,would prove Fer-mat’s Last Theorem.This occupies more than16pages of the manuscript,

in very…ne,detailed,polished writing.We analyze Germain’s plan in this section,ending with a comparison between Manuscripts A and D and what

it suggests.

2.1Germain’s plan for proving Fermat’s Last Theorem

Let us call Germain’s condition on auxiliary primes =2Np+1

Condition N-C(Non-Consecutivity).There are n o two nonzero c onsecutive p th power residues,modulo .

Early on in Manuscript A,Germain states that for each…xed N(except when N is a multiple of3;for which she shows that Condition N-C always fails8),there will only be…nitely many exceptional numbers p for which the auxiliary =2Np+1will fail to satisfy Condition N-C(only primes of the form =2Np+1can possibly satisfy the N-C condition9).Much of her manuscript is devoted to supporting this claim;while not carried to fruition, Germain’s insight was vindicated much later when proven by E.Wendt in 1894[Di,Ri,We].10

Establishing Condition N-C for each N,and an induction on N

In order to establish Condition N-C for various N and p,Germain engages

in extensive analysis over many pages of the general consequences of nonzero consecutive p-th power residues modulo a prime =2Np+1(N never a multiple of3).Her analysis actually encompasses all natural numbers for p,not just primes.This is important in relation to the form of ,since she intends to carry out a mathematical induction on N,and eventually explains in detail her ideas about how the induction should go.She employs throughout the notion and notation of congruences introduced by Gauss,and utilizes to great e¤ect a keen understanding that the2Np multiplicative units mod are cyclic,generated by a primitive2Np-th root of unity,enabling

her to engage in detailed analyses of the relative placement of the nonzero

p-th powers(i.e.,the2N-th roots of1)amongst the residues.She is acutely 8See[Ri,p.127].

9See[Ri,p.124].

10Germain’s aim follows immediately from Wendt’s recasting of the condition in terms

of a circulant determinant depending on N.Condition N-C fails to hold for only if

p divides the determinant,which is nonzero for all N not divisible by3.There is no indication that Wendt was aware of Germain’s work.

14

aware that subgroups of the group of units are also cyclic,and of their orders and interrelationships,and uses this in a detailed way.Throughout her analyses she deduces that in may instances the existence of nonzero consecutive p-th power residues would ultimately force2to be a p-th power mod ,and she therefore repeatedly concludes that Condition N-C holds under the hypothesis that we will call

Condition2-N-p(2is Not a p-th power).The number2is not a p-th power residue,modulo .

Notice that this hypothesis is always a necessary condition for Condition N-C to hold,since if2is a p-th power,then obviously1and2are nonzero consecutive p-th powers;so making this assumption is no restriction,and Germain is simply exploring whether2-N-p is also su￠cient to ensure N-C.

Always assuming this hypothesis,which we shall discuss later,and also the always necessary condition that3-N,Germain’s analysis initially shows that if there exist two nonzero consecutive p-th power residues,then by inverting them,or subtracting them from 1,or iterating combinations of these transformations,she can obtain more pairs of nonzero consecutive p-th power residues.11

Germain proves that,under her constant assumption that2is not a p-th power residue modulo ,this transformation process will produce at least6 completely disjoint such pairs,i.e.,involving at least12actual p-th power residues.Therefore since there are precisely2N nonzero p-th power residues modulo ,she instantly proves Condition N-C for all auxiliary primes with N=1;2;4;5as long as p satis…es Condition2-N-p.Germain continues with more detailed analysis of these permuted pairs of consecutive p-th power residues(still assuming Condition2-N-p)to verify Condition N-C for N=7 (excluding p=2)and N=8(here she begins to use inductive information for earlier values of N).

At this point Germain explains her general plan to continue the method of analysis to higher N,and how she would use induction on N for all p simultaneously.In a nutshell,she argues that the existence of nonzero consecutive p-th power residues would have to result in a pair of such,x; 11In fact these transformations are permuting the pairs of consecutive residues according to an underlying group with six elements,which we shall discuss later.Germain even notes,when explaining the situation in her letter to Gauss[Ge1],that from any one of the six pairs,her transformations will reproduce the…ve others.This approaches the existence of inverses in a group,and Germain’s phenomenon,if it had become known, could have served as one of several important examples in the early nineteenth century that stimulated the development of the group concept.

15

x+1,for which x is(congruent to)an odd power(necessarily less than2N) of x+1.She claims that one must then analyze cases of expansions of the binomial,depending on the value of N,to arrive at the desired contradiction, and she carries out a complete detailed calculation for N=10(excluding p=2;3)as a speci…c“example”of how she says the induction will work in general.

We have found it quite di￠cult to understand fully this part of the manuscript.Germain’s claims may in fact hold,but we have not managed to verify them completely from what she says.We have di￠culty with an aspect of her argument for N=7,with her explanation of exactly how her mathematical induction will proceed,and with an aspect of her explanation of how in general a pair x;x+1with the property claimed above is ensured. Finally,Germain’s example calculation for N=10is much more ad hoc than one would like as an illustration of how things would go in a mathematical induction on N.Nonetheless,her instincts here were correct,as proven by Wendt.

The interplay between N and p

But lest the reader think that proving N-C for all N,each with…nitely many excepted p,would immediately solve the Fermat problem,note that what is actually needed,for each…xed prime p,is that Condition N-C holds for in…nitely many N,not the other way around.For instance,perhaps p=3 must be excluded from the validation of Condition N-C for all su￠ciently large N,in which case Germain’s method would not prove Fermat’s Last Theorem for p=3.Germain makes it clear early in the manuscript that she recognizes this issue,that her results do not completely resolve it,and that she has not proved Fermat’s claim for a single predetermined exponent.But she also states that she strongly believes that the needed requirements do in fact hold,and that her results for N 10strongly support this.Indeed, note that so far the only odd prime excluded in any veri…cations was p=3 for N=10(recall,though,that we have not yet examined Condition2-N-p,which must also hold in all her arguments,and which will also exclude certain combinations of N and p when it fails).

Germain’s…nal comment on this issue states…rst that as one proceeds to ever higher values of N,there is always no more than a“very small number”of values of p for which Condition N-C fails.If indeed this,the very crux of the whole approach,were the case,in particular if the number of such excluded p were bounded uniformly,say by K,for all N,which is what she in e¤ect claims,then a little re?ection reveals that indeed her

16

method would have proven Fermat’s Last Theorem for all but K values of

p,although one would not necessarily know which values.She herself states

that this would prove the theorem for in…nitely many p,even though not for

a single predetermined value of p.It is in this sense that Germain believed

her method could prove in…nitely many instances of Fermat’s Last Theorem. Verifying Condition2-N-p

We conclude our exposition of Germain’s grand plan in Manuscript A with

her analysis of Condition2-N-p,which was required throughout all her argu-

ments above.She points out that for2to be a p-th power mod =2Np+1

means that22N 1(mod )(since the multiplicative structure is cyclic). Clearly for…xed N this can only occur for…nitely many p,and she easily

determines these exceptional cases through N=10,simply by calculating

and factoring each22N 1by hand,and observing whether any of the prime factors are of the form2Np+1for any natural number p.To illustrate,for N=7she writes that

214 1=3 43 127=3 (14 3+1) (14 9+1);

so that p=3;9are the only values for which Condition2-N-p fails for this

N.

Germain then presents a summary table of all her results verifying Con-

dition N-C for auxiliary primes using relevant values of N 10and primes

2

in the table are impressive.Aside from the case of =43=14 3+1just

illustrated,the only other auxiliary primes in the range of her table which

must be omitted are =31=10 3+1,which she determines fails Condition

2-N-p,and =61=20 3+1,which was an exception in her N-C analysis

for N=10.In fact each N in her table ends up having at least…ve primes

p with2

N-C condition.

While the number of p requiring exclusion for Condition2-N-p may ap-

pear“small”for each N,there seems no obvious reason why it should nec-

essarily be uniformly bounded for all N;Germain does not discuss this issue

speci…cally for Condition2-N-p.As indicated above,without such a bound it 12The table is slightly?awed in that she includes =43=14 3+1for N=7despite the excluding calculation we just illustrated,which Germain herself had just written out; it thus seems that the manuscript may have simple errors,suggesting it may sadly never have received good criticism from another mathematician.

17

is not clear that this method could actually prove any instances of Fermat’s theorem.

Results of the grand plan

To summarize,Germain had a sophisticated and highly developed grand plan for proving Fermat’s Last Theorem for in…nitely many exponents.It relied heavily on expertise with the multiplicative structure in a cyclic prime …eld and a set(group)of transformations of consecutive p-th powers,and it involved many clever ideas which we have not laid out here in detail.She carried her program out in an impressive range of values for the necessary auxiliary primes,believed that the evidence indicated one could push it fur-ther using mathematical induction by her methods,and she was optimistic that by doing so it would prove Fermat’s Last Theorem for in…nitely many prime exponents.In hindsight we know that,promising as it may have seemed at the time,the program can never be carried to completion.

2.2Did Germain ever know her grand plan cannot succeed? To answer this question we examine the published record,correspondence with Gauss,and a letter from Germain to Legendre.

Libri claims that such a plan cannot work

Published indication that Germain’s method can not succeed in proving Fer-mat’s Last Theorem came in work of Guglielmo(Guillaume)Libri,a rising mathematical star in the1820s.It is a bit hard to track and compare the content of his relevant works and their dates,partly because Libri presented or published several di¤erent works all with the same title,but some of these were also multiply published.Our interest is in the content of just two dif-ferent works.In1829Libri published a set of his own memoirs[Li1].One of these is titled Mémoire sur la théorie des nombres,republished later word for word as three papers in Crelle’s Journal[Li].The memoir published in 1829ends by applying Libri’s study of the number of solutions of various congruence equations to the situation of Fermat’s Last Theorem.Among other things,Libri shows that for exponents3and4,there can be at most …nitely many auxiliary primes satisfying the N-C condition.And he claims that his methods will clearly show the same for all higher exponents.Libri explicitly notes that his result proves that the attempts of others to prove Fermat’s Last Theorem by…nding in…nitely many such auxiliaries are in vain.

18

Libri also writes in his1829memoir that all the results he obtains were already presented in two earlier memoirs of1823and1825to the Academy of Sciences in Paris.Libri’s1825presentation to the Academy was also published,in1833/1838[Li3],confusingly with the same title as the1829 memoir.This presumably earlier document13is quite similar to the publi-cation of1829,in that it develops methods for determining the number of solutions to quite general congruence equations,including that of the N-C condition,but it does not explicitly work out the details for the N-C condi-tion applying to Fermat’s Last Theorem,as did the1829memoir.Thus it seems that close followers of the Academy should have been aware by1825 that Libri’s work would doom the auxiliary prime approach to Fermat’s Last Theorem,but it is hard to pin down exact dates.Much later,P.Pepin[Pe1, pp.318–319][Pe2]and A.-E.Pellet[Pe,p.93](see[Di][Ri,pp.292–293]) con…rmed all of Libri’s claims,and L.E.Dickson[Di1,Di2]gave speci…c bounds.For completeness,we mention that Libri also published a memoir on number theory in1820,his very…rst publication,with the title Memo-ria Sopra La Teoria Dei Numeri[Li2],but it was much shorter and does not contain the same type of study or results on the number of solutions to congruence equations.

What Germain knew and when:Gauss,Legendre,and Libri

So did Germain ever know from Libri or otherwise that her grand plan to prove Fermat’s Last Theorem could not work,and if so,when?

We know that in1819she was enthusiastic in her letter to Gauss about her method for proving Fermat’s Last Theorem,based on extensive work ex-empli…ed by Manuscript A.14In it Germain details several speci…c examples of her results on the N-C condition that match perfectly with Manuscript A,and which she explicitly explains have been extracted from an already much older note(“d’une note dejáancienne”)that she has not had the time 13One can wonder when the document…rst published in1833,but based on Libri’s1825 Academy presentation,was really written or…nalized.Remarks he makes in it suggest,

though,that it was essentially his1825presentation.

14Near the end she even expresses to Gauss how a brand new work by L.Poinsot[Po] will help her further her e¤orts to con…rm the N-C condition by giving a new way of

working with the p-th powers mod =2Np+1.She interpets them as the solutions of

the binomial equation of degree2N,i.e.,of x2N 1=0.Poinsot’s memoir takes the point of view that the mod solutions of this equation can be obtained by…rst considering the

equation over the complex numbers,where much was already known about the complex

2N-th roots of unity,and then considering these roots as mod p integers by replacing the p 1by an integer whose square yields 1mod p.

complex number

19

to recheck.In fact everything in the extensive letter to Gauss matches the details of Manuscript A.This suggests that Manuscript A is likely the older note in question,and considerably predates her1819letter to Gauss.Thus 1819is our lower bound for the answer to our question.We also know that by1823Legendre had written his memoir crediting Germain with her the-orem,but without even mentioning the method of…nding in…nitely many auxiliary primes that Germain had pioneered to try to prove Fermat’s Last Theorem.We know,too,that Germain wrote notes in1822on Libri’s1820 memoir,15but this…rst memoir did not study modular equations,hence was not relevant for the N-C condition.It seems likely that she came to know of Libri’s claims dooming her method,based either on his presentations to the Academy in1823/25or the later memoir published in1829,particularly because Germain and Libri had met and were personal friends from1825, as well as frequent correspondents.It thus seems probable that sometime between1819and1823or1825Germain would have come to realize that her grand plan could not work.

Germain proves to Legendre that the plan fails for p=3

In fact,though,we do not need to speculate about Germain’s knowledge of Libri’s work in order to answer our primary question,since we have found separate evidence of Germain’s realization that her method of proving Fer-mat’s Last Theorem cannot succeed,at least not in all cases.While Man-uscript A and her letter of1819to Gauss evince her belief that for every prime p>2,there will be in…nitely many auxiliary primes satisfying the N-C condition,there is an undated letter to Legendre in which Germain actually proves the opposite for p=3[Ge4].Although we have found noth-ing else in the way of correspondence between Legendre and Germain on Fermat’s Last Theorem,we are fortunate to know of this one critical letter, held in the Samuel Ward papers of the New York Public Library.16 Sophie Germain began her three page letter by thanking Legendre for “telling”her“yesterday”that one can prove that all numbers of the form 6a+1larger than13have a pair of consecutive(nonzero)cubic residues.This amounts to saying that for p=3,no auxiliary primes of the form =2Np+1 15Germain’s three pages of notes[Ge5,cass.7,ins.56][Ce,p.233],while not directly about Fermat’s Last Theorem,do indicate an interest in modular solutions of roots of unity equations,which is what encompasses the distribution of p-th powers modulo .Compare this with what she wrote to Gauss about Poinsot’s work,discussed in the previous footnote. 16The Samuel Ward papers include“letters by famous mathematicians and scientists acquired by Ward with his purchase of the library of mathematician A.M.Legendre.”We thank Louis Bucciarelli for providing us with this lead.

20

图表与口诀记忆when、as、while的区别

图表与口诀记忆when、as、while的区别 1.图表与口诀前知识 关键是比较主从句子的动词，看其动词的持续性。瞬间的理解成点，持续的理解成线。主从关系有：点（点点、点线），线线，线点。 点：为瞬间动词，准确地称为“终止性动词”，指动词具有某种内在界限的含义，一旦达到这个界限，该动作就完成了。如come（来），一旦“到来”，该动作就不再继续下去了。 瞬间动词：arrive, begin, borrow, become, buy, catch, come, die, find, go，give, graduate, join, kill, lose, leave, marry, realize… 线：为非瞬间动词，准确地称为叫“延续性动词”。包括动态动词静态动词。 动态动词：live, sit, stand, study, talk, work, write… 静态动词（状态动词）：情感、看法、愿望等。Be, belong, consist, exist, feel, hate, have, hope, love, want… 兼有瞬时和非瞬时的动词：feel，look，move，run，work，write…，需要根据不同的语境判断。 2. when、as、while的区别一览表 【表格说明】：第一个点或者线表示从句谓语动词的持续性特征，黑点表示从句所表示的动作持续短，为瞬间动词，线表示持续长，为非瞬间动词。1~7为主句与从句所表示的动作时间有重合，第8为主句与从句所表示的动作不是同时发生，而是有先后顺序。 线线重相并发生， 长线” 【主句谓语为非瞬间动词中的 动态动词】 【记忆：等线动， 相并发生，但： 【主句谓语为非瞬间动词中的 静态动词】 【记忆：等线动，

再探翻译质量评估参数 何三宁

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when,while,as的区别

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意大利 2 2 - 澳大利亚 2 - 4 全部 女子跻身巴黎奥运殿堂 世界名城巴黎，是法国首都，位于巴黎盆地中央，横跨塞纳河两岸，水陆交通方便，是举行国际比赛的理想地方。第二届奥运会于1900年5月20曰（一说5月14曰）至10月28曰在这里举行。 在1894年巴黎国际体育大会上，顾拜旦曾建议第一届奥运会于1900年与世界博览会同时在巴黎举行，借以扩大奥运会的影响。但是，雅典奥运会胜利举行后，希腊人对奥运会表现了极大的热情，想推翻第二届会址设在巴黎的决议。希腊一些有影响的人士认为，奥运会是希腊民族文化的部分，它只能在希腊举行，雅典应成为奥运会的固定地址。如果易地他国召开，则是对伟大的、光辉灿烂的希腊文化的公开掠夺。但是，当时已接替维凯拉斯任国际奥委会主席的顾拜旦在这个问题上坚持不让。他说，奥林匹克运动是希腊

的，也是全世界的。他认为奥运会必须在不同国家举行，才能使之具有国际性和富有生命力。希腊终于被说服，巴黎赢得了主办权。 但是，顾拜旦想利用世界博览会来扩大奥林匹克运动影响的打算，却未能如愿以偿。法国政府对博览会的兴趣远胜于奥运会，而承办两项会务的主要负责人阿夫雷德·皮卡尔，是一个不热心体育的人，他对顾拜旦提出的奥运会筹备方案淡然置之，甚至不屑一顾。他把主要精力放在博览会上，而对奥运会的比赛项目，曰程、场地等均无周密安排，更谈不上花费巨款去兴建体育设施。出生于巴黎贵族家庭的顾拜旦，曾经十分失望地在曰记中写到：”世界上有一个对奥运会非常冷淡的地方，这就是巴黎。”1900年5月14曰，作为博览会附属的第二届奥运会总算开始了。组织者根本就没想到要举行开幕式。所以，场面冷冷清清。尽管如此，还是有与第一届奥运会所无法比拟之处。巴黎繁华的城市、便利的交通、可供游览的名胜古迹等等，十分具有吸引力。参加本届奥运会的国家由上届的13个增加到22个（一说21个），运动员达到1330人，参赛人数也比上届多得多。东道主派出了由884名运动员组成的庞大代表团，人数居首位。首次参赛的有比利时、波希米亚、海地、西班牙、意大利、加拿大、古巴、荷兰、挪威和印度。印度选手诺尔曼·普理查德

英文翻译版质量管理制度及职责

赫克力士上海技术中心项目质量管理制度及职责 ., . 15, 2007

目录 1、项目经理岗位职责() (3) 2、现场经理岗位职责： 3、项目技术负责人岗位职责： (3) 4、施工员岗位职责： (4) 5、技术员岗位职责： (5) 6、质检员岗位职责： (5) 7、材料员岗位职责： (7) 8、资料员岗位职责： (7) 9、操作工人岗位职责 (8) 10、现场质量管理制度 (9) 10.1、工程项目总承包责任制度 (9) 10.2 技术交底制度 (15) 10.3、材料进场检验制度 (16) 10.4、样板引路制度 (20) 10.5、施工挂牌制度 (22) 10.6、过程三检制度 (24) 10.7、质量否决制度 (28) 10.8、成品保护制度 (31) 10.9、工程质量评定、验收制度 (33) 10.10、竣工服务承诺制度 (36) 11、培训上岗制度 (37) 12、工程质量事故报告及调查制度 (38) 13、现场材料的存放与管理制度 (39) 14、现场设备的存放与管理制度 (42) 1、项目经理岗位职责()

1.1向公司经理负责，全面负责项目各项管理工作，是项目总负责人，接受公司各职能部门的监督和指导；, , . 1.2是项目质量和安全的第一责任人，对本工程的质量和安全全面负责，明确项目经理部各部门的质量和安全责任，并督促其履行职责；, . ’, . 1.3负责组织实施公司的质量体系文件，组织制定并实施项目质量管理措施及各分部分项工程质量检验评定；. ’ . 1.4履行公司质量体系文件规定的各项职责；’s . 1.5负责项目资金的全盘控制及运作； 1.6全盘控制项目成本，定期组织项目成本的核算；, . 1.7负责分供方的选择、签约及参与工程承包合同的评审工作。, . 2、现场经理岗位职责： 2.1协助项目经理搞好项目各项管理工作，对工程的质量、安全、进度、文明施工等进行全盘控制；, ’s , , . 2.2负责与发包人、监理单位、设计人及上级行政主管部门的对接及有关协调工作；, , 2.3贯彻上级主管部门有关施工生产计划、指令、文件，主持编制月生产计划及资源需用计划，监督并督促有关部门实施；’s , , , . 2.4协助项目经理搞好项目成本控制，主持项目成本核算工作；, . 2.5参与分供方的选择、签约及工程承包合同的评审工作；, . 2.6协助项目经理履行公司质量体系文件规定的各项职责； . 2.7协助项目经理组织实施公司的质量体系文件、组织制定并实施项目质量管理措施及各分部分项工程质量检验评定；, . 2.8定期主持召开项目经理部生产例会，定期组织并主持项目分包商及发包人指定分包商的项目协调会；; ’s . 2.9主持重大安全方案的编制并组织重大安全事故的处理工作，定期组织项目安全检查；, ; . 2.10组织并主持项目重大质量事故的处理工作，主持项目项目竣工验收；; . 2.11负责对内对外函件的批示，并检查其落实情况。, . 3、项目技术负责人岗位职责： 3.1协助项目经理组织实施公司质量体系程序规定的各项职责，督促项目施工技术人员履行其质量职责, . 3.2组织编制施工组织设计及项目质量保证计划，负责组织编制特殊过程和专题施工方案及作业指导书，检查督促施工组织设计、项目质量保证计划的实施，参与分部分项工程质量检验评定；, , , , ; . 3.3协助项目经理对项目质量进行控制、管理和督促；, . 3.4负责施工过程中质量、技术问题的处理； .

第七--when-while-as-区别及练习.

When while as区别 一、根据从句动作的持续性来区分 1、“主短从长”型：即主句是一个短暂性动作，而从句是一个持续性动作，此时三者都可用。如： Jim hurt his arm while［when， as］ he was playing tennis. 吉姆打网球时把手臂扭伤了。 2、“主长从长”型：即主句和从句为两个同时进行的动作或存在的状态，且强调主句动作或状态延续到从句所指的整个时间，此时通常要用while。 I always listen to the radio while I’m driving. 我总是一边开车一边听收音机。 He didn’t ask me in; he kept me standing at the door while he read the message. 他没有让我进去，他只顾看那张条子，让我站在门口等着。 但是，若主句和从句所表示的两个同时进行的动作含有“一边……一边”之意时，则习惯上要用as。如： He swung his arms as he walked. 他走路时摆动着手臂。 3、“主长从短”型：即主句是一个持续性动作，而从句是一个短暂性动作，此时可以用as 或when，但不能用while。如： It was raining hard when [as] we arrived. 我们到达时正下着大雨。 二、根据主句与从句动作是否同时发生来区分 1、若主句与从句表示的是两个同时发生的短暂性动作，含有类似汉语“一……就”的意思，英语一般要用as (也可用when)。如： The ice cracked as [when] I stepped onto it. 我一踩冰就裂了。 2、若主句与从句表示的是两个几乎同时发生的短暂性动作，含有类似汉语“刚要……就”“正要……却”的意思，英语一般要用as(也可用when)，且此时通常连用副词just。如： I caught him just when [as] he was leaving the building. 他正要离开大楼的时候，我把他截住了。 三、根据是否具有伴随变化来区分 若要表示主句动作伴随从句动作同时发展变化，有类似汉语“随着”的意思，英语习惯上要用as，而不用when或while。如： The room grew colder as the fire burnt down. 随着炉火逐渐减弱，房间越来越冷。 注：若不是引导从句，而是引出一个短语，则用with，不用as。如： With winter coming on, it’s time to buy warm clothes. 随着冬天到来，该买暖和衣裳了。 四、根据从句动作的规律性来区分 若暗示一种规律性，表示“每当……的时候”，英语一般要用when。如： It’s cold when it snows. 下雪时天冷。 五、根据主从句动作的先后顺序来区分 若主句与从句所表示的动作不是同时发生，而是有先后顺序时，一般要用when。

3笔翻译考核标准

下面转载一个译文审核标准： （一次转不完，分两份哈） 译文审核标准 Elanso后台编辑将根据以下的标准来评定译文是否通过审核，一方面是要强化译员的品质意识、将空泛的各派翻译标准极具操作性地落到实处，另一方面是有助于甄别翻译质量的高低、提高译文的质量，让更多的读者喜欢您的译文。 此审核标准，集合了多位专家评审的自身学习体会和工作经验，全面参照英国皇家特许语言学家学会（Chartered Institute of Linguists）和澳大利亚翻译资格认可局（National Accreditation Authority for Translators and Interpreters）的笔译评价标准以及中华人民共和国国家标准《翻译服务规范第1部分：笔译》。 《译文审核标准》提出“内容完整”、“信息准确”、“表达恰当”、“语言品质”、“遵从惯例”、“风格贴近”、“技术细节”以及“整体效果”等八个方面的质量指标，基本涵盖了对翻译的所有常见要求。如果一篇译文，译文中如果不符合该标准中的任何3项，译文都将被编辑审核不通过。 一、内容完整Completeness 是否提交完整的译文、提交时间是否不超出约定期限，这是翻译质量的默认衡量标准。 1. 翻译完备completeness 2. 交稿及时punctuality 二、信息准确Accuracy 提交的译文是否完全准确地传达了原文信息，没有添加、删减、扭曲、错乱，这是翻译质量的首要衡量标准。 1. 信息添加addition 原文：The UN released the document in The Hague and in more than 100 nations’ capitals, including Beijing. 译文：这份报告是联合国对包括北京在内的一百多个国家的首都进行调查后，在海牙公布的。 n 原文并未提及联合国对各国首都进行了调查，显然这是译者没有深究原文意思，先入为主地胡编乱造的。实际上，原文把“The Hague”和“capitals”并列区别开来，一是因为联合国的这份文件很可能是在海牙（国际法院所在地）首发的，二是因为海牙并非荷兰的首都，无法纳入“capitals”。 改译：联合国在海牙和包括北京在内的一百多个国家的首都发布了该报告。 2. 信息删减omission 原文：The Engineer shall be at liberty to object to and require the Contractor to remove forthwith from the Works any person employed by the Contractor in or about the execution or maintenance of the Works who, in the opinion of the Engineer, misconducts himself or is incompetent or negligent in the proper performance of his duties or whose employment is otherwise considered by the Engineer to be undesirable and such person shall not be again employed upon the Works without the written permission of the Engineer.

国际组织名称中英文对照及缩写教程文件

国际组织名称中英文对照及缩写

国际组织名称中英文对照及缩写 African Development Fund 非洲开发基金 ADF African Groundnut Council 非洲花生理事会 AGC African Liberation Committee 非洲解放委员会 African Postal Union 非洲邮政联盟 AFPU African Postal and Telecommunication Union 非洲邮政电信联盟 APU/ATU African Timber Organization 非洲木材组织 ATO Agency for the Protection of Number weapons in Latin America 拉丁美洲禁止核武器组织OPANAL American International Chamber of Commerce 美国国际商会 Andean Pact Organization 安第斯条约组织（安第斯集团） Andean Group Andean Reserve Fund 安第斯储备基金会 FAR Anzus Council 澳新美理事会 Arab African International Bank 阿拉伯-非洲国际银行 Arab Common Market 阿拉伯共同市场 Arab Fund for Economic and Social Development 阿拉伯经济和社会发展基金会 AFESD Arab Inter-Parliamentary Union 阿拉伯议会联盟 AIPU Arab-Latin American Bank 阿拉伯-拉丁美洲银行 ARLABANK Arab Monetary Fund 阿拉伯货币基金组织 AMF Arab Organization for Standardization and Metrology 阿拉伯标准化和计量组织 ASMO Arab States Broadcasting Union 阿拉伯国家广播联盟 ASBU Arab Summit Conference 阿拉伯国家首脑会议 ASEAN Regional Forum 东盟地区论坛 ARF Asian-African Conference（Bandung Conference）亚非会议（万隆会议）

When while as的区别和用法(综合整理)

When while as的区别和用法 when的用法 当主句使用持续性动词时. Dave was eating,when the doorbell rang.门铃响时,大卫在吃饭. 2.一个动作紧接着另一个动作发生. When the lights went out, I lit some candles.灯灭了,我赶紧点上一些蜡烛. 3.谈论生命中的某一阶段,或过去的某段时间. His mother called him Robbie when he was a baby. 在他很小时,他妈妈叫他Robbin. 4.指"每一次" When I turn on the TV, smoke comes out the back. 每当我打开电视,就有烟从后面冒出. while/as 的用法 从句多为进行时,而且为持续性动词. I'll look after the children while you are making dinner. 你做饭,我来照顾孩子. 注意事项: (1) “主短从长”型：主句表示的是一个短暂性动作，从句表示的是一个持续性动作，三者都可用： He fell asleep when [while, as] he was reading. 他看书时睡着了。 Jim hurt his arm while［when，as］he was playing tennis. 吉姆打网球时把手臂扭伤了。 As［When，While］she was waiting for the train，she became very impatient. 她在等火车时，变得很不耐烦。 (2) “主长从长”型：若主、从句表示两个同时进行的持续性动作，且强调主句表示的动作延续到从句所指的整个时间，通常要用while： Don’t talk while you’re eating. 吃饭时不要说话。 I kept silent while he was writing. 在他写的时候，我默不做声。 但是，若主从句表示的两个同时进行的动作含有“一边…一边”之意思，通常用as：

2020巴黎第六大学世界排名

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120个研究单位，分布于33个地点。有学生3万人，博士生3千5 百人。有五个学院。分别是化学学院、工程学院、数学学院、医学院、物理学院、生命科学学院、地球与环境科学学院以及生物多样 性学院。 巴黎第六大学有以下学术部门：化学、工程、数学、医学、物理学、生命科学与地球科学、环境以及生物多样性。附属于巴黎第六 大学的机构还有PolytechParis工程师学院、巴黎天体物理学研究 所以及儒勒·昂利·庞加莱研究所。儒勒·昂利·庞加莱研究所专 攻数学和理论物理学。校历以学期为基础，主要教学语言为法语， 部分研究生课程提供英语教学。学士课程通常为三年，硕士课程为 二年。学校大约有一百多科研实验室。这些实验室归属于四大科研 领域：模拟与工程;能量、物质与宇宙;地球与环境;以及生命健康。 学校有三个海洋观测站，用于研究海洋学、海洋生物学及相关领域。 巴黎第六大学皮埃尔和玛丽居里两位著名校友名字命名，共有诺贝尔奖得主17人。除了皮埃尔和玛丽居里，其他与巴黎第六大学有 联系的诺贝尔奖得主还包括HIV病毒发现者弗朗索瓦丝?巴尔-西诺西、原子研究先驱让·巴蒂斯特·皮兰以及量子物理学家克劳德?科恩?坦诺奇。 巴黎第六大学享有国际声誉。其3万4千名学生中，约20%属于 国际生。它是欧洲三大创新网路的积极成员。它们分别是ClimateKIC、EITICTLabs和EITHealth。(欧洲共有五大创新网路。) 除此之外，巴黎第六大学也倡导海外学习，有国际学士学位4个，国际硕士学位16个，国际博士学位4个，同世界各地的大学有合作。 巴黎第六大学有120个实验室，得到3750名研究员和教授支持。每年，其出版物数量约为8千5百，其核心目标之一是应对21世纪 社会当前面临的问题。 巴黎第六大学位于巴黎市中心，是一个令人振奋的留学目的地。学校为学生提供可操控的时间表，让学生有时间参加巴黎的俱乐部、体育活动，餐馆巴黎的各种文化景点。

GHTF—质量管理体系--过程验证指南中文版

GHRF/SG3/N99-10:2004 (第2版) 最终文件 标题：质量管理体系——过程确认指南 编写：GHTF 第3研究组 签署：全球协调任务组织 日期：2004年1月第2版 Taisuke Hojo, GHTF主席 本文件由全球协调任务组织制作，该组织是一个志愿团体，由医疗器械管理机 构和管理行业的代表组成。本文件着重为管理机构提供关于医疗器械法规使用方面的非约束性指导，其撰写是经过多方面征求意见的。 本文件的印制、发售或使用是不受限制的。但是，将本文件部分或全部引用到其它文件，或将它翻译成英语以外的其它语言，均不代表全球协调任务组织认同。

GHTF第3研究组—质量管理体系 过程确认指南— 2004年1月 第2页 过程确认指南 目录 0前言 (3) 1 目的和范围 (5) 1.1目的 (5) 1.2 范围 (5) 2 定义 (5) 3 质量管理体系范围内的过程确认 (5) 3.1 过程确认的判定 (6) 3.2 举例 (7) 4 过程确认的统计方法和工具 (8) 5 确认的实施 (8) 5.1 准备阶段 (8) 5.2 方案编制 (9) 5.3 安装鉴定（IQ） (10) 5.4 操作鉴定（OQ） (10) 5.5 性能鉴定（PQ） (11) 5.6 最终报告 (12) 6 确认状态的保持 (12) 6.1 监视和控制 (12) 6.2 过程和（或）产品的改变 (12) 6.3 连续的控制状态 (12) 6.4 再确认原因举例 (12) 7 过程确认中历史数据的使用 (13) 8 活动小结 (13) 附录 A 过程确认的统计方法和工具 (15) B 确认的举例 (25)

when,while,as引导时间状语从句的区别

when，while，as引导时间状语从句的区别 when，while，as显然都可以引导时间状语从句，但用法区别非常大。 一、when可以和延续性动词连用，也可以和短暂性动词连用；而while和as只能和延续性动词连用。 ①Why do you want a new job when youve got such a good one already？（get 为短暂性动词）你已经找到如此好的工作，为何还想再找新的？ ②Sorry，I was out when you called me．（call为短暂性动词）对不起，你打电话时我刚好外出了。 ③Strike while the iron is hot．（is为延续性动词，表示一种持续的状态）趁热打铁。 ④The students took notes as they listened．（listen为延续性动词）学生们边听课边做笔记。 二、when从句的谓语动词可以在主句谓语动作之前、之后或同时发生；while 和as从句的谓语动作必须是和主句谓语动作同时发生。 1．从句动作在主句动作前发生，只用when。 ①When he had finished his homework，he took a short rest．（finished先发生）当他完成作业后，他休息了一会儿。 ②When I got to the airport，the guests had left．（got to后发生）当我赶到飞机场时，客人们已经离开了。 2．从句动作和主句动作同时发生，且从句动作为延续性动词时，when，while，as都可使用。 ①When ／While ／As we were dancing，a stranger came in．（dance为延续性动词）当我们跳舞时，一位陌生人走了进来。 ②When ／While ／As she was making a phonecall，I was writing a letter．（make为延续性动词）当她在打电话时，我正在写信。 3．当主句、从句动作同时进行，从句动作的时间概念淡化，而主要表示主句动作发生的背景或条件时，只能用as。这时，as常表示“随着……”；“一边……，一边……”之意。 ①As the time went on，the weather got worse．（as表示“随着……”之意） ②The atmosphere gets thinner and thinner as the height increases．随着高度的增加，大气越来越稀薄。 ③As years go by，China is getting stronger and richer．随着时间一年一年过去，中国变得越来越富强了。 ④The little girls sang as they went．小姑娘们一边走，一边唱。 ⑤The sad mother sat on the roadside，shouting as she was crying．伤心的妈妈坐在路边，边哭边叫。 4．在将来时从句中，常用when，且从句须用一般时代替将来时。 ①You shall borrow the book when I have finished reading it．在我读完这本书后，你可以借阅。 ②When the manager comes here for a visit next week，Ill talk with him about this．下周，经理来这参观时，我会和他谈谈此事。 三、when用于表示“一……就……”的句型中（指过去的事情）。 sb．had hardly（＝scarcely）done sth．when．．．＝Hardly ／Scarcely had sb．done sth．when．．．