Fermat's little theorem



Fermat's little theorem

(Michel Waldschmidt on Fermat)

A note of late March 2017 concerning the opening of the Maple worksheet below, indeed my Maple worksheets in general. You may encounter a problem opening my Maple worksheets (as indeed I do myself) here at my web site - it would appear to depend on the internet browser being used. Thus, if I attempt to open one of my worksheets using Interner Explorer there is never a problem, whereas if I use Firefox (or Google Chrome) then all that one sees - this is just an example - is something like this: {VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 0 128 0 128 1 0 1 0 0 1 0 0 0 0 } {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 128 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" and so on (almost at infinitum)...

I asked Maplesoft for advice on this and they recommended doing the following (and I found it worked):

  1. Don't attempt directly to open the worksheet (by clicking on my link), instead right click and save the worksheet to (say) the download folder.

  2. Now that the worksheet is in the download folder (cut it out if you wish and put into whatever folder you wish) you may open the worksheet in the usual way (I should add that while I see this 'works', I have absolutely no idea as to why it does... ) [END OF NOTE].


Friday 17th August 2001 was the 400th anniversary of the birth of Pierre de Fermat, and by way of a personal homage I decided that for the ICTMT5 meeting in Klagenfurt, Austria - held the week prior to Fermat's anniversary - I would offer a talk, using Maple, called Fermat's little theorem. It was never my intention to cover all of my prepared talk in Klagenfurt, and, in the event, I covered less than 0.1% of what I actually prepared. My Maple worksheet (129KB) may be downloaded here, and a (large) html version of it may be downloaded here (large because Maple converts all outputs to gif files, and there are 447 of those in the worksheet). 

I dedicated my lecture to Mark Daly - a former colleague, and friend - as a token of my regard for him.

Klaus Barner of Kassel university, Germany, disputes the date of Fermat's birth, and interested persons ought to read his papers:

Pierre de Fermat (1601? - 1665), European Mathematics Society Newsletter No. 42, December 2001

How old did Fermat become? 2001

for a well argued case that "Fermat was most probably born in 1607."

Some weeks before I went to Klagenfurt I posted a question to the moderated Number Theory Mailing List to try determine who first called Fermat's little theorem 'little.' The responses (some private, some public) of that question may be read below:

My NT mail of 20th June:

Dear colleagues,

By was of a personal homage to Fermat, and to mark the 400th anniversary of his birth (this coming August 17th. Are the French postal authorities bringing out a special stamp?), I am giving a talk - "Fermat's 'little' theorem", using Maple - at the ICTMT5 Conference in Klagenfurt, Austria, in the week before the anniversary. In time I will put the lecture up at my web site in mws and html formats (they will contain much, much more than I can cover in a 30 minute talk).

Could someone please answer this banal question: who (and when) first used the 'little'? Dickson doesn't use 'little' in his monumental History of the Theory of Numbers (its Chapter III, Vol I, has title "Fermat's and Wilson's theorems"); Hardy and Wright don't use 'little' (at least not my 4th edition copy); etc

I have a suspicion (I don't know why) that it was E.T. Bell, but I would very much appreciate - just for the sake of accuracy - an exact placing. Thanking you, John Cosgrave

A private reply: Dear John, I am replying just to you. I am essentially certain that it was common German practice to refer to FLT as the 'great' theorem of Fermat. A secondary reference to that is Paulo Ribenboim's translation of Wolfskehl's will. I imagine that therefore they too used the adjective little in respect of Fermat's Little Theorem. Were I lecturing on the matter I would likely declare that no doubt Klein already referred to the Little Theorem. English usage in the matter is surely just a translation of german usage.

Another private reply: John, In Carmichael's 1914 number theory text, he says "it is often referred to as the simple Fermat theorem." He also attributes Euler's theorem to Fermat. He says later that Fermat had written down the case for primes, and didn't prove it. Euler gave the proof of what we call Euler's theorem, yet he attributes it still to Fermat. In Landau's 1927 elementary number theory book, he calls what we call Euler's theorem "the so-called Little Fermat Theorem".

Edwin Clark #1: A search of math journals up to 1950 for fermat's little theorem OR little fermat theorem on JSTOR yields: MacLane uses "little Fermat Theorem" in Modular Fields, Saunders Mac Lane, American Mathematical Monthly, Vol. 47, No. 5. (May, 1940), pp. 259-274. Kaplansky uses "Fermat's "little theorem" " in Lucas's Tests for Mersenne Numbers, Irving Kaplansky American Mathematical Monthly, Vol. 52, No. 4. (Apr., 1945), pp. 188-190.
Edwin Clark

A private reply to EC from Germany which EC forwarded to me: Dear Edwin, as is usual in such situation, I do not have a substantial answer to your query right off the cuff. However, I do recall that ever since I got into contact with the German literature on basic algebra or number theory, that is since the time of my T"ubingen studies in the years after 1952, the terminology of the "little Fermat" and the "big Fermat" was absolutely current in a way as if this had been so for generations down the line. ("Kleiner Fermatscher Satz" versus "Grosser Fermatscher Satz".) My tedacher G"unter Pickert used this terminology in his algebra book of 1951, and I am sure that van der Waerden has it in his algebra book of 1934 (or so). I believe that Saunders MacLane got his doctorate 1934 in G"ottingen. I would imagine that the terminology had been long current at that time. Klein had died in G"ottingen in 1925 but the time of his mathematical and pedagogical influence dates back to the 19th century. If I were a betting man I would bet that this terminology also goes back to the great number theoreticians of the nineteenth century like Kummer and Dedekind. I shall keep my eyes open and perk my ears. Should I find something, I shall let you know.

And another from the same German source: Yes, Edwin, I confirm that the German editions accessible in the Darmstadt library just refer to "Satz von Fermat". (I am just a little bit surprised, because I had indeed speculated that this very influential book had the terminology of "kleiner Fermatscher Satz." The next source to check is the enormously influential algebra book by Weber (Einf"uhrung in die Algebra) [Heinrich Weber, 1843--1913 taught Hilbert in K"onigsberg]. The first Volume appeared 1894. I ckecked the second edition of 1898 and he DOES speak of the "Lehrsatz von Fermat" (no "klein" here anywhere). Also in the second volume where the theorem reappears it is called "Satz von Fermat". I am glad that I did not bet on the appearance of "kleiner Fermatscher Satz" in 19th century literature. Weber would certainly reflect terminology that was current in the 19th century. Now your hypothesis that the terminology was generated some time in the last century (I mean the 20th) perhaps G"ottingen gains credence. Helmut Hasses Textbooks beginning 1927 at the latest repeatedly speak of "Kleiner Fermatscher Satz". Pickert was Hasses's student. So that explains where HIS terminology would come from. This could also explain Saunders' using this terminology; he could easily have picked up this terminology in G"ottingen. I have no explanation why van der Waerden does NOT use it; he studied in G"ottingen hanging out, among others with my teacher Hellmuth Kneser who was Courant's assistant having gotten his Ph.D. under Hilbert. I have no recollection how Hellmuth Kneser spoke of this theorem. Incidentally, in his "Zahlbericht" Hasse speaks of "Fermats Vermutung" and obviously not of "Fermats Grossem Satz"

A less scholarly reply: I often joke to my students that it was his wife that called it that.....it always goes over well.

A reply from Brazil: Morris Kline (Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972, p. 276) is speaking about the minor and major theorems of Fermat. Emile Borel (les Nombres Premiers, PUF, QSJ 571, 1958) is speaking about the theorem of Fermat and the last theorem of Fermat. Idem for Rouse Ball and Coxeter (Mathematical Recreations and Essays, Univ. of Toronto Press, 12th edition, 1974). Idem for O. Ore (Number Theory and its History, McGraw-Hill, 1948).

Another private reply: Dear John Cosgrave: There is no mention of "little" in Bell's book Men of Mathematics (1937) or in his Development of Mathematics (1945). I have a copy of Uspensky & Heaslet, Elementary Number Theory, published in 1939, and on p. 145 you will find: 'Fermat states a result of which an important theorem, now known as the "little Fermat theorem," is a consequence.' This indicates to me that someone had coined the term "little Fermat theorem" by the time Uspensky & Heaslet published their book in 1939.

In 1936, D. H. Lehmer wrote a famous paper in the American Mathematical Monthly (v. 43, pp. 347-354) entitled "On the converse of Fermat's theorem," but the word "little" does not appear in this paper. It is my opinion (as a student of D. H. Lehmer) that he would certainly have used the term "little" if it was commonly used at that time.

I did check Kaplansky's 1945 paper in the Monthly (v. 52, pp. 188-190)entitled "On Lucas' test for Mersenne numbers" and on p. 140 he refers to Fermat's "little theorem."

So this little bit of detective work reveals that the term "little Fermat theorem" probably first appeared sometime between 1936 and 1939 and was in common usage by 1945.

A mail to the Historia Mathematica mailing list:

----- Original Message -----
From: "Maryvonne Spiesser" <[email protected]>
To: <[email protected]>
Sent: Friday, June 29, 2001 10:33 AM
Subject: Re: [HM] stamps about maths

Dear listmembers, A special stamp will be brought out on the occasion of the 400th anniversary of Fermat's supposed birthday. "First day" will hold on 18th of August in Beaumont de Lomagne. Envelopes, postcards, ... will be buyable during the conference "Fermat, 400 ans apres" in Toulouse, University of Toulouse III, October 18th-19th 2001. M. Spiesser


Delivery-date: Tue, 26 Jun 2001 12:31:20 +0200
From: Peter Flor <[email protected]>
Subject: Re: [HM] Fwd: origin of 'little' in Fermat's 'little' theorem
To: [email protected]
Reply-To: [email protected]

There are much older sources for this usage than those quoted so far. Here is one: Kurt Hensel, "Zahlentheorie" (1913): "F"ur jede endliche Gruppe besteht nun ein Fundamentalsatz, welcher der kleine Fermatsche Satz genannt zu werden pflegt, weil ein ganz spezieller Teil desselben zuerst von Fermat bewiesen worden ist." . Translation: "There is a fundamental theorem holding in every finite group, usually called Fermat's little Theorem because Fermat was the first to have proved a very special part of it." So at least in German, this seems to have been a term in common use, in 1913. Best wishes to all, Peter.

Contact details. jbcosgrave at gmail.com