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
FLT.
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.
Conjectures
-----------
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
classes)
(a) conjectures arising from Diophantine approximation theory such
as the ABC conjecture, the Szpiro conjecture, the Hall conjecture,
etc.
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
conjecture:
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
theorem].
(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.
References:
[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
Hall problem). MANY OTHER "MATHEMATICAL" GAMES ARE EXPLAINED
IN THE REC.PUZZLES FAQ. READ IT BEFORE ASKING IN SCI.MATH.
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 archive-request@questrel.com
an email message consisting of the text
send switch
Also any other FAQ list can be obtained through anonymous ftp from
rtfm.mit.edu.
References
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.
References:
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.
References:
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]
[ ]
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Questions and Answers _Compiled_ by:
Alex Lopez-Ortiz alopez-o@maytag.UWaterloo.ca
Department of Computer Science University of Waterloo
Waterloo, Ontario Canada