Fermat's *little* theorem

(Michel
Waldschmidt on Fermat)

Friday 17^{th} August 2001 was the 400^{th} 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:

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

Added Wed. 29^{th} August. Jonathan Borwein has kindly informed me of the
following web supplement to the MAA Monthly paper on Giuga's conjecture that I reference
in my Fermat talk:

http://142.58.12.69/personal/jborwein/mathcamp00.html

(Note added Feb. 2011. In 2007 I made two research visits (two weeks in May, three weeks in November) to Dalhousie university to work with my co-author Karl Dilcher. On both occasions I stayed with Jon and Judi Borwein; happy days.)

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 400^{th} 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" <spiesser@cict.fr>

To: <historia-matematica-digest@chasque.apc.org>

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

**Finally**:

Delivery-date: Tue, 26 Jun 2001 12:31:20 +0200

From: Peter Flor <peter.flor@kfunigraz.ac.at>

Subject: Re: [HM] Fwd: origin of 'little' in Fermat's 'little' theorem

To: historia-matematica@chasque.apc.org

Reply-To: historia-matematica@chasque.apc.org

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.