Math FAQ- update for 6 July 1993

1Q.- Fermat's Last Theorem, status of .. #
11Q.- There are three doors, The Monty Hall problem, Master Mind and
      other games .. #
19Q.- Erdos Number #
20Q.- Odd Perfect Number #
21Q.- Why is there no Nobel in mathematics? #
22Q.- General References and textbooks... #

1Q:  What is the current status of Fermat's last theorem?
    (There are no positive integers x,y,z, and n > 2 such that
    x^n + y^n = z^n)
    I heard that <insert name here> claimed to have proved it but later
    on the proof was found to be wrong. ...
    (wlog we assume x,y,z to be relatively prime)

A:  The status of FLT has remained remarkably constant.  Every few
    years, someone claims to have a proof ... but oh, wait, not quite.
    Meanwhile, it is proved true for ever greater values of the exponent
    (but not all of them), and ties are shown between it and other
    conjectures (if only we could prove one of them), and ... so it has
    been for quite some time.  It has been proved that for each
    exponent, there are at most a finite number of counter-examples to

    Here is a brief survey of the status of FLT. It is not intended to
    be 'deep', but it is rather for non-specialists.

    The theorem is broken into 2 cases. The first case assumes
    (abc,n) = 1. The second case is the general case.

    What has been PROVED

    First Case.

    It has been proven true up to 7.568x10^17 by the work of Wagstaff &
    Tanner, Granville&Monagan, and Coppersmith. They all used extensions
    of the Wiefrich criteria and improved upon work performed by
    Gunderson and Shanks&Williams.

    The first case has been proven to be true for an infinite number of
    exponents by Adelman, Frey, et. al. using a generalization of the
    Sophie Germain criterion

    Second Case:

    It has been proven true up to n = 150,000 by Tanner & Wagstaff. The
    work used new techniques for computing Bernoulli numbers mod p and
    improved upon work of Vandiver. The work involved computing the
    irregular primes up to 150,000. FLT is true for all regular primes
    by a theorem of Kummer. In the case of irregular primes, some
    additional computations are needed.

    UPDATE :

    Fermat's Last Theorem has been proved true up to exponent 4,000,000
    in the general case. The method used was essentially that of Wagstaff:
    enumerating and eliminating irregular primes by Bernoulli number
    computations. The computations were performed on a set of NeXT
    computers by Richard Crandall et al.

    Since the genus of the curve a^n + b^n = 1, is greater than or equal
    to 2 for n > 3, it follows from Mordell's theorem [proved by
    Faltings], that for any given n, there are at most a finite number
    of solutions.


    There are many open conjectures that imply FLT. These conjectures
    come from different directions, but can be basically broken into
    several classes: (and there are interrelationships between the

    (a) conjectures arising from Diophantine approximation theory such
    as the ABC conjecture, the Szpiro conjecture, the Hall conjecture,

    For an excellent survey article on these subjects see the article
    by Serge Lang in the Bulletin of the AMS, July 1990 entitled
    "Old and new conjectured diophantine inequalities".

    Masser and Osterle formulated the following known as the ABC

    Given epsilon > 0, there exists a number C(epsilon) such that for
    any set of non-zero, relatively prime integers a,b,c such that
    a+b = c we have

    max( |a|, |b|, |c|) <= C(epsilon) N(abc)^(1 + epsilon)

    where N(x) is the product of the distinct primes dividing x.

    It is easy to see that it implies FLT asymptotically. The conjecture
    was motivated by a theorem, due to Mason that essentially says the
    ABC conjecture IS true for polynomials.

    The ABC conjecture also implies Szpiro's conjecture [and vice-versa]
    and Hall's conjecture. These results are all generally believed to
    be true.

    There is a generalization of the ABC conjecture [by Vojta] which is
    too technical to discuss but involves heights of points on
    non-singular algebraic varieties . Vojta's conjecture also implies
    Mordell's theorem [already known to be true]. There are also a
    number of inter-twined conjectures involving heights on elliptic
    curves that are related to much of this stuff. For a more complete
    discussion, see Lang's article.

    (b) conjectures arising from the study of elliptic curves and
    modular forms. -- The Taniyama-Weil-Shmimura conjecture.

    There is a very important and well known conjecture known as the
    Taniyama-Weil-Shimura conjecture that concerns elliptic curves.
    This conjecture has been shown by the work of Frey, Serre, Ribet,
    et. al. to imply FLT uniformly, not just asymptotically as with the
    ABC conj.

    The conjecture basically states that all elliptic curves can be
    parameterized in terms of modular forms.

    There is new work on the arithmetic of elliptic curves. Sha, the
    Tate-Shafarevich group on elliptic curves of rank 0 or 1. By the way
    an interesting aspect of this work is that there is a close
    connection between Sha, and some of the classical work on FLT. For
    example, there is a classical proof that uses infinite descent to
    prove FLT for n = 4. It can be shown that there is an elliptic curve
    associated with FLT and that for n=4, Sha is trivial. It can also be
    shown that in the cases where Sha is non-trivial, that
    infinite-descent arguments do not work; that in some sense 'Sha
    blocks the descent'. Somewhat more technically, Sha is an
    obstruction to the local-global principle [e.g. the Hasse-Minkowski

    (c) Conjectures arising from some conjectured inequalities involving
    Chern classes and some other deep results/conjectures in arithmetic
    algebraic geometry.

    I can't describe these results since I don't know the math. Contact
    Barry Mazur [or Serre, or Faltings, or Ribet, or ...]. Actually the
    set of people who DO understand this stuff is fairly small.

    The diophantine and elliptic curve conjectures all involve deep
    properties of integers. Until these conjecture were tied to FLT,
    FLT had been regarded by most mathematicians as an isolated problem;
    a curiosity. Now it can be seen that it follows from some deep and
    fundamental properties of the integers. [not yet proven but
    generally believed].

    This synopsis is quite brief. A full survey would run to many pages.


    [1] J.P.Butler, R.E.Crandall, & R.W.Sompolski
    "Irregular Primes to One Million"
     Math. Comp. 59 (October 1992) pp. 717-722

    H.M. Edwards, Fermat's Last Theorem, A Genetic Introduction to
    Algebraic Number Theory, Springer Verlag, New York, 1977

    P. Ribenboim, Thirteen Lectures on Fermat's Last Theorem,
    Springer Verlag, New York, 1979

    Number Theory Related to Fermat's Last Theorem, Neal Koblitz, editor,
    Birkh\"auser Boston, Inc., 1982, ISBN 3-7643-3104-6

11Q:  There are three doors, and there is a car hidden behind one
    of them, Master Mind and other games ..

A:  Read frequently asked questions from rec.puzzles, where the
    problem is solved and carefully explained. (The Monty

    Your chance of winning is 2/3 if you switch and 1/3 if you don't.
    For a full explanation from the frequently asked questions list
    for rec.puzzles, send to the address
    an email message consisting of the text

               send switch

    Also any other FAQ list can be obtained through anonymous ftp from


    American Mathematical Monthly, January 1992.

    For the game of Master Mind it has been proven that no more than
    five moves are required in the worst case. For references look at

    One such algorithm was published in the Journal of Recreational
    Mathematics; in '70 or '71 (I think), which always solved the
    4 peg problem in 5 moves. Knuth later published an algorithm which
    solves the problem in a shorter # of moves - on average - but can
    take six guesses on certain combinations.

    Donald E. Knuth, The Computer as Master Mind, J. Recreational Mathematics
    9 (1976-77), 1-6.

19Q:  What is the Erdos Number?

     Form an undirected graph where the vertices are academics, and an
     edge connects academic X to academic Y if X has written a paper
     with Y.  The Erdos number of X is the length of the shortest path
     in this graph connecting X with Erdos.

     What is the Erdos Number of X ? for a few selected X in {Math,physics}

     Erdos has Erdos number 0.  Co-authors of Erdos have Erdos number 1.
     Einstein has Erdos number 2, since he wrote a paper with Ernst Straus,
     and Straus wrote many papers with Erdos.

     Why people care about it?

     Nobody seems to have a reasonable answer...

     Who is Paul Erdos?

     Paul Erdos is an Hungarian mathematician, he obtained his PhD
     from the University of Manchester and has spent most of his
     efforts tackling "small" problems and conjectures related to
     graph theory, combinatorics, geometry and number theory.

     He is one of the most prolific publishers of papers; and is
     also and indefatigable traveller.


      Caspar Goffman, And what is your Erdos number?, American Mathematical
      Monthly v. 76 (1969), p. 791.

20Q:  Does there exist a number that is perfect and odd?

     A given number is perfect if it is equal to the sum of all its proper
     divisors. This question was first posed by Euclid in ancient Greece.
     This question is still open.  Euler proved that if  N  is an odd
     perfect number, then in the prime power decomposition of N, exactly
     one exponent is congruent to 1 mod 4 and all the other exponents are
     even. Furthermore, the prime occurring to an odd power must itself be
     congruent to 1 mod 4.  A sketch of the proof appears in Exercise 87,
     page 203 of Underwood Dudley's Elementary Number Theory, 2nd ed.
     It has been shown that there are no odd perfect numbers < 10^300.

21Q.- Why is there no Nobel in mathematics? #

     Nobel prizes were created by the will of Alfred Nobel, a notable
     swedish chemist.

     One of the most common --and unfounded-- reasons as to why Nobel
     decided against a Nobel prize in math is that [a woman he proposed
     to/his wife/his mistress] [rejected him beacuse of/cheated him
     with] a famous mathematician. Gosta Mittag-Leffler is often claimed
     to be the guilty party.

     There is no historical evidence to support the story.

     For one, Mr. Nobel was never married.

     There are more credible reasons as to why there is no Nobel prize
     in math. Chiefly among them is simply the fact he didn't care much
     for mathematics, and that it was not considered a practical
     science from which humanity could benefit (a chief purpose
     for creating the Nobel Foundation).

     Here are some relevant facts:

     1. Nobel never married, hence no ``wife". (He did have a mistress,
     a Viennese woman named Sophie Hess.)

     2. Gosta Mittag-Leffler was an important mathematician in Sweden
     in the late 19th-early 20th century.  He was the founder of the
     journal Acta Mathematica, played an important role in helping the
     career of Sonya Kovalevskaya, and was eventually head of the
     Stockholm Hogskola, a technical institute. However, it seems
     highly unlikely that he would have been a leading candidate for
     an early Nobel Prize in mathematics, had there been one -- there
     were guys like Poincare and Hilbert around, after all.

     3.  There is no evidence that Mittag-Leffler
     had much contact with Alfred Nobel (who resided in Paris
     during the latter part of his life), still less that there was
     animosity between them for whatever reason.  To the contrary,
     towards the end of Nobel's life Mittag-Leffler was engaged in
     ``diplomatic" negotiations to try to persuade Nobel to designate
     a substantial part of his fortune to the Hogskola. It seems hardly
     likely that he would have undertaken this if there was prior
     bad blood between them.  Although initially Nobel seems to have
     intended to do this, eventually he came up with the Nobel Prize
     idea -- much to the disappointment of the Hogskola, not to mention
     Nobel's relatives and Fraulein Hess.

     According to the very interesting study by Elisabeth Crawford,
     ``The Beginnings of the Nobel Institution", Cambridge Univ. Press,
     1984, pages 52-53:

     ``Although it is not known how those in responsible positions
     at the Hogskola came to believe that a *large* bequest was
     forthcoming, this indeed was the expectation, and the
     disappointment was keen when it was announced early in 1897 that
     the Hogskola had been left out of Nobel's final will in 1895.
     Recriminations followed, with both Pettersson and Arrhenius
     [academic rivals of Mittag-Leffler in the administration of the
     Hogskola] letting it be known that Nobel's dislike for
     Mittag-Leffler had brought about what Pettersson termed the
     `Nobel Flop'.  This is only of interest because it may have
     contributed to the myth that Nobel had planned to institute a prize
     in mathematics but had refrained because of his antipathy to
     Mittag-Leffler or --in another version of the same story-- because
     of their rivalry for the affections of a woman...."

     4.  A final speculation concerning the psychological element.
     Would Nobel, sitting down to draw up his testament, presumably
     in a mood of great benevolence to mankind, have allowed a mere
     personal grudge to distort his idealistic plans for the monument
     he would leave behind?
     Nobel, an inventor and industrialist, did not create a prize in
     mathematics simply because he was not particularly interested
     in mathematics or theoretical science.  His will speaks of
     prizes for those ``inventions or discoveries" of greatest
     practical benefit to mankind.  (Probably as a result of this
     language, the physics prize has been awarded for experimental work
     much more often than for advances in theory.)

     However, the story of some rivalry over a woman is obviously
     much more amusing, and that's why it will probably continue to
     be repeated.


     Mathematical Intelligencer, vol. 7 (3), 1985, p. 74.

     Elisabeth Crawford, ``The Beginnings of the Nobel Institution",
     Cambridge Univ. Press, 1984.

22Q.- General References and textbooks... #

     [a list of general references and most commonly used textbooks]
     [                                                             ]

Questions and Answers _Compiled_ by:

Alex Lopez-Ortiz                    
Department of Computer Science                      University of Waterloo
Waterloo, Ontario                                                   Canada