1193 lines
48 KiB
Plaintext
1193 lines
48 KiB
Plaintext
Newsgroups: sci.math,news.answers
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From: alopez-o@maytag.uwaterloo.ca (Alex Lopez-Ortiz)
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Subject: sci.math: Frequently Asked Questions
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Message-ID: <C17J4M.JLq@watdragon.uwaterloo.ca>
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Summary: (version 3.8)
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Organization: University of Waterloo
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Date: Thu, 21 Jan 1993 14:05:10 GMT
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Lines: 1183
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Archive-Name: sci-math-faq
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Version: $Id: sci-math-faq,v 3.8 92/12/26 18:45:00 $
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This is a list of Frequently Asked Questions for sci.math (version 3.8).
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Any contributions/suggestions/corrections are most welcome. Please use
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* e-mail * on any comment concerning the FAQ list.
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Changes and additions are marked with a # on the table of contents.
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This FAQ list (and most others, for that matter) is available via anonymous
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ftp at rtfm.mit.edu (18.172.1.27).
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The list of contributors to this FAQ list is too large to include here;
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but thanks are due to all of them (you know who you are folks).
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Table of Contents
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-----------------
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1Q.- Fermat's Last Theorem, status of .. #
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2Q.- Four Colour Theorem, proof of ..
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3Q.- Values of Record Numbers
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4Q.- General Netiquette
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5Q.- Computer Algebra Systems, application of ..
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6Q.- Computer Algebra Systems, references to ..
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7Q.- Fields Medal, general info ..
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8Q.- 0^0=1. A comprehensive approach
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9Q.- 0.999... = 1. Properties of the real numbers ..
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10Q.- Digits of Pi, computation and references
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11Q.- There are three doors, The Monty Hall problem, Master Mind and
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other games .. #
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12Q.- Surface and Volume of the n-ball
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13Q.- f(x)^f(x)=x, name of the function ..
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14Q.- Projective plane of order 10 ..
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15Q.- How to compute day of week of a given date
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16Q.- Axiom of Choice and/or Continuum Hypothesis?
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17Q.- Cutting a sphere into pieces of larger volume
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18Q.- Pointers to Quaternions
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19Q.- Erdos Number #
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20Q.- Odd Perfect Number #
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21Q.- Why is there no Nobel in mathematics? #
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22Q.- General References and textbooks... #
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1Q: What is the current status of Fermat's last theorem?
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(There are no positive integers x,y,z, and n > 2 such that
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x^n + y^n = z^n)
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I heard that <insert name here> claimed to have proved it but later
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on the proof was found to be wrong. ...
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(wlog we assume x,y,z to be relatively prime)
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A: The status of FLT has remained remarkably constant. Every few
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years, someone claims to have a proof ... but oh, wait, not quite.
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Meanwhile, it is proved true for ever greater values of the exponent
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(but not all of them), and ties are shown between it and other
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conjectures (if only we could prove one of them), and ... so it has
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been for quite some time. It has been proved that for each
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exponent, there are at most a finite number of counter-examples to
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FLT.
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Here is a brief survey of the status of FLT. It is not intended to
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be 'deep', but it is rather for non-specialists.
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The theorem is broken into 2 cases. The first case assumes
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(abc,n) = 1. The second case is the general case.
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What has been PROVED
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--------------------
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First Case.
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It has been proven true up to 7.568x10^17 by the work of Wagstaff &
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Tanner, Granville&Monagan, and Coppersmith. They all used extensions
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of the Wiefrich criteria and improved upon work performed by
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Gunderson and Shanks&Williams.
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The first case has been proven to be true for an infinite number of
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exponents by Adelman, Frey, et. al. using a generalization of the
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Sophie Germain criterion
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Second Case:
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It has been proven true up to n = 150,000 by Tanner & Wagstaff. The
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work used new techniques for computing Bernoulli numbers mod p and
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improved upon work of Vandiver. The work involved computing the
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irregular primes up to 150,000. FLT is true for all regular primes
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by a theorem of Kummer. In the case of irregular primes, some
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additional computations are needed.
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UPDATE :
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Fermat's Last Theorem has been proved true up to exponent 4,000,000
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in the general case. The method used was essentially that of Wagstaff:
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enumerating and eliminating irregular primes by Bernoulli number
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computations. The computations were performed on a set of NeXT
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computers by Richard Crandall et al.
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Since the genus of the curve a^n + b^n = 1, is greater than or equal
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to 2 for n > 3, it follows from Mordell's theorem [proved by
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Faltings], that for any given n, there are at most a finite number
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of solutions.
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Conjectures
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-----------
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There are many open conjectures that imply FLT. These conjectures
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come from different directions, but can be basically broken into
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several classes: (and there are interrelationships between the
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classes)
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(a) conjectures arising from Diophantine approximation theory such
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as the ABC conjecture, the Szpiro conjecture, the Hall conjecture,
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etc.
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For an excellent survey article on these subjects see the article
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by Serge Lang in the Bulletin of the AMS, July 1990 entitled
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"Old and new conjectured diophantine inequalities".
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Masser and Osterle formulated the following known as the ABC
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conjecture:
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Given epsilon > 0, there exists a number C(epsilon) such that for
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any set of non-zero, relatively prime integers a,b,c such that
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a+b = c we have
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max( |a|, |b|, |c|) <= C(epsilon) N(abc)^(1 + epsilon)
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where N(x) is the product of the distinct primes dividing x.
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It is easy to see that it implies FLT asymptotically. The conjecture
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was motivated by a theorem, due to Mason that essentially says the
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ABC conjecture IS true for polynomials.
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The ABC conjecture also implies Szpiro's conjecture [and vice-versa]
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and Hall's conjecture. These results are all generally believed to
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be true.
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There is a generalization of the ABC conjecture [by Vojta] which is
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too technical to discuss but involves heights of points on
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non-singular algebraic varieties . Vojta's conjecture also implies
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Mordell's theorem [already known to be true]. There are also a
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number of inter-twined conjectures involving heights on elliptic
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curves that are related to much of this stuff. For a more complete
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discussion, see Lang's article.
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(b) conjectures arising from the study of elliptic curves and
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modular forms. -- The Taniyama-Weil-Shmimura conjecture.
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There is a very important and well known conjecture known as the
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Taniyama-Weil-Shimura conjecture that concerns elliptic curves.
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This conjecture has been shown by the work of Frey, Serre, Ribet,
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et. al. to imply FLT uniformly, not just asymptotically as with the
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ABC conj.
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The conjecture basically states that all elliptic curves can be
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parameterized in terms of modular forms.
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There is new work on the arithmetic of elliptic curves. Sha, the
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Tate-Shafarevich group on elliptic curves of rank 0 or 1. By the way
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an interesting aspect of this work is that there is a close
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connection between Sha, and some of the classical work on FLT. For
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example, there is a classical proof that uses infinite descent to
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prove FLT for n = 4. It can be shown that there is an elliptic curve
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associated with FLT and that for n=4, Sha is trivial. It can also be
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shown that in the cases where Sha is non-trivial, that
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infinite-descent arguments do not work; that in some sense 'Sha
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blocks the descent'. Somewhat more technically, Sha is an
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obstruction to the local-global principle [e.g. the Hasse-Minkowski
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theorem].
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(c) Conjectures arising from some conjectured inequalities involving
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Chern classes and some other deep results/conjectures in arithmetic
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algebraic geometry.
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I can't describe these results since I don't know the math. Contact
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Barry Mazur [or Serre, or Faltings, or Ribet, or ...]. Actually the
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set of people who DO understand this stuff is fairly small.
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The diophantine and elliptic curve conjectures all involve deep
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properties of integers. Until these conjecture were tied to FLT,
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FLT had been regarded by most mathematicians as an isolated problem;
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a curiosity. Now it can be seen that it follows from some deep and
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fundamental properties of the integers. [not yet proven but
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generally believed].
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This synopsis is quite brief. A full survey would run to many pages.
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References:
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[1] J.P.Butler, R.E.Crandall, & R.W.Sompolski
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"Irregular Primes to One Million"
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Math. Comp. 59 (October 1992) pp. 717-722
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H.M. Edwards, Fermat's Last Theorem, A Genetic Introduction to
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Algebraic Number Theory, Springer Verlag, New York, 1977
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P. Ribenboim, Thirteen Lectures on Fermat's Last Theorem,
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Springer Verlag, New York, 1979
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Number Theory Related to Fermat's Last Theorem, Neal Koblitz, editor,
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Birkh\"auser Boston, Inc., 1982, ISBN 3-7643-3104-6
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2Q: Has the Four Colour Theorem been solved?
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(Every planar map with regions of simple borders can be coloured
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with 4 colours in such a way that no two regions sharing a non-zero
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length border have the same colour.)
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A: This theorem was proved with the aid of a computer in 1976.
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The proof shows that if aprox. 1,936 basic forms of maps
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can be coloured with four colours, then any given map can be
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coloured with four colours. A computer program coloured this
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basic forms. So far nobody has been able to prove it without
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using a computer. In principle it is possible to emulate the
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computer proof by hand computations.
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References:
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K. Appel and W. Haken, Every planar map is four colourable,
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Bulletin of the American Mathematical Society, vol. 82, 1976
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pp.711-712.
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K. Appel and W. Haken, Every planar map is four colourable,
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Illinois Journal of Mathematics, vol. 21, 1977, pp. 429-567.
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T. Saaty and Paul Kainen, The Four Colour Theorem: Assault and
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Conquest, McGraw-Hill, 1977. Reprinted by Dover Publications 1986.
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K. Appel and W. Haken, Every Planar Map is Four Colorable,
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Contemporary Mathematics, vol. 98, American Mathematical Society,
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1989, pp.741.
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F. Bernhart, Math Reviews. 91m:05007, Dec. 1991. (Review of Appel
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and Haken's book).
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3Q: What are the values of:
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largest known Mersenne prime?
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A: It is 2^756839-1. It was discovered by a Cray-2 in England in 1992.
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It has 227,832 digits.
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largest known prime?
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A: The largest known prime is the Mersenne prime described above.
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The previous record holder, and the largest known non-Mersenne prime,
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is 391581*2^216193-1. See Brown, Noll, Parady, Smith, Smith, and
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Zarantonello, Letter to the editor, American Mathematical Monthly,
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vol. 97, 1990, p. 214. Throughout history, the largest known prime
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has almost always been a Mersenne prime; the period between Brown
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et al's discovery in Aug 1989 and Slowinski & Gage's in March 1992
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is one of the few exceptions.
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largest known twin primes?
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A: The largest known twin primes are 1706595*2^11235 +- 1.
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See B. K. Parady and J. F. Smith and S. E. Zarantonello,
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Smith, Noll and Brown.
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Largest known twin primes, Mathematics of Computation,
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vol.55, 1990, pp. 381-382.
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largest Fermat number with known factorization?
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A: F_11 = (2^(2^11)) + 1 which was factored by Brent & Morain in
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1988. F9 = (2^(2^9)) + 1 = 2^512 + 1 was factored by
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A.K. Lenstra, H.W. Lenstra Jr., M.S. Manasse & J.M. Pollard
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in 1990. The factorization for F10 is NOT known.
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Are there good algorithms to factor a given integer?
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A: There are several that have subexponential estimated
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running time, to mention just a few:
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Continued fraction algorithm,
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Class group method,
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Quadratic sieve algorithm,
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Elliptic curve algorithm,
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Number field sieve,
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Dixon's random squares algorithm,
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Valle's two-thirds algorithm,
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Seysen's class group algorithm,
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A.K. Lenstra, H.W. Lenstra Jr., "Algorithms in Number Theory",
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in: J. van Leeuwen (ed.), Handbook of Theoretical Computer
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Science, Volume A: Algorithms and Complexity, Elsevier, pp.
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673-715, 1990.
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List of record numbers?
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A: Chris Caldwell maintains "THE LARGEST KNOWN PRIMES (ALL KNOWN
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PRIMES WITH 2000 OR MORE DIGITS)"-list. Send him mail to
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bf04@UTMartn.bitnet (preferred) or kvax@utkvx.UTK.edu, on any new
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gigantic primes (greater than 10,000 digits), titanic primes
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(greater than 1000 digits).
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What is the current status on Mersenne primes?
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A: Mersenne primes are primes of the form 2^p-1. For 2^p-1 to be prime
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we must have that p is prime. The following Mersenne primes are
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known.
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nr p year by
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-----------------------------------------------------------------
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1-5 2,3,5,7,13 in or before the middle ages
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6-7 17,19 1588 Cataldi
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8 31 1750 Euler
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9 61 1883 Pervouchine
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10 89 1911 Powers
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11 107 1914 Powers
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12 127 1876 Lucas
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13-14 521,607 1952 Robinson
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15-17 1279,2203,2281 1952 Lehmer
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18 3217 1957 Riesel
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19-20 4253,4423 1961 Hurwitz & Selfridge
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21-23 9689,9941,11213 1963 Gillies
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24 19937 1971 Tuckerman
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25 21701 1978 Noll & Nickel
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26 23209 1979 Noll
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27 44497 1979 Slowinski & Nelson
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28 86243 1982 Slowinski
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29 110503 1988 Colquitt & Welsh jr.
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30 132049 1983 Slowinski
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31 216091 1985 Slowinski
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32? 756839 1992 Slowinski & Gage
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The way to determine if 2^p-1 is prime is to use the Lucas-Lehmer
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test:
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Lucas_Lehmer_Test(p):
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u := 4
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for i from 3 to p do
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u := u^2-2 mod 2^p-1
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od
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if u == 0 then
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2^p-1 is prime
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else
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2^p-1 is composite
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fi
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The following ranges have been checked completely:
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2 - 355K and 430K - 520K
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More on Mersenne primes and the Lucas-Lehmer test can be found in:
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G.H. Hardy, E.M. Wright, An introduction to the theory of numbers,
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fifth edition, 1979, pp. 16, 223-225.
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(Please send updates to alopez-o@maytag.UWaterloo.ca)
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4Q: I think I proved <insert big conjecture>. OR
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I think I have a bright new idea.
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What should I do?
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A: Are you an expert in the area? If not, please ask first local
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gurus for pointers to related work (the "distribution" field
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may serve well for this purposes). If after reading them you still
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think your *proof is correct*/*idea is new* then send it to the net.
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5Q: I have this complicated symbolic problem (most likely
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a symbolic integral or a DE system) that I can't solve.
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What should I do?
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A: Find a friend with access to a computer algebra system
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like MAPLE, MACSYMA or MATHEMATICA and ask her/him to solve it.
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If packages cannot solve it, then (and only then) ask the net.
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6Q: Where can I get <Symbolic Computation Package>?
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This is not a comprehensive list. There are other Computer Algebra
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packages available that may better suit your needs. There is also
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a FAQ list in the group sci.math.symbolics. It includes a much larger
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list of vendors and developers. (The FAQ list can be obtained from
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rtfm.mit.edu via anonymous ftp).
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A: Maple
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Purpose: Symbolic and numeric computation, mathematical
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programming, and mathematical visualization.
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Contact: Waterloo Maple Software,
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160 Columbia Street West,
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Waterloo, Ontario, Canada N2L 3L3
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Phone: (519) 747-2373
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wmsi@daisy.uwaterloo.ca wmsi@daisy.waterloo.edu
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A: DOE-Macsyma
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Purpose: Symbolic and mathematical manipulations.
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Contact: National Energy Software Center
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Argonne National Laboratory 9700 South Cass Avenue
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Argonne, Illinois 60439
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Phone: (708) 972-7250
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A: Pari
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Purpose: Number-theoretic computations and simple numerical
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analysis.
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Available for Sun 3, Sun 4, generic 32-bit Unix, and
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Macintosh II. This is a free package, available by ftp from
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math.ucla.edu (128.97.64.16).
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Contact: questions about pari can be sent to pari@mizar.greco-prog.fr
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A: Mathematica
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Purpose: Mathematical computation and visualization,
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symbolic programming.
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Contact: Wolfram Research, Inc.
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100 Trade Center Drive Champaign,
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IL 61820-7237
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Phone: 1-800-441-MATH
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A: Macsyma
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Purpose: Symbolic and mathematical manipulations.
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Contact: Symbolics, Inc.
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8 New England Executive Park East
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Burlington, Massachusetts 01803
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United States of America
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(617) 221-1250
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macsyma@Symbolics.COM
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A: Matlab
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Purpose: `matrix laboratory' for tasks involving
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matrices, graphics and general numerical computation.
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Contact: The MathWorks, Inc.
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21 Eliot Street
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South Natick, MA 01760
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508-653-1415
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info@mathworks.com
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A: Cayley
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Purpose: Computation in algebraic and combinatorial structures
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such as groups, rings, fields, modules and graphs.
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Available for: SUN 3, SUN 4, IBM running AIX or VM, DEC VMS, others
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Contact: Computational Algebra Group
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University of Sydney
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NSW 2006
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Australia
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Phone: (61) (02) 692 3338
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Fax: (61) (02) 692 4534
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cayley@maths.su.oz.au
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7Q: Let P be a property about the Fields Medal. Is P(x) true?
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A: There are a few gaps in the list. If you know any of the
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missing information (or if you notice any mistakes),
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please send me e-mail.
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Year Name Birthplace Age Institution
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---- ---- ---------- --- -----------
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1936 Ahlfors, Lars Helsinki Finland 29 Harvard U USA
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1936 Douglas, Jesse New York NY USA 39 MIT USA
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1950 Schwartz, Laurent Paris France 35 U of Nancy France
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1950 Selberg, Atle Langesund Norway 33 Adv.Std.Princeton USA
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1954 Kodaira, Kunihiko Tokyo Japan 39 Princeton U USA
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1954 Serre, Jean-Pierre Bages France 27 College de France France
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1958 Roth, Klaus Breslau Germany 32 U of London UK
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1958 Thom, Rene Montbeliard France 35 U of Strasbourg France
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1962 Hormander, Lars Mjallby Sweden 31 U of Stockholm Sweden
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1962 Milnor, John Orange NJ USA 31 Princeton U USA
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1966 Atiyah, Michael London UK 37 Oxford U UK
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1966 Cohen, Paul Long Branch NJ USA 32 Stanford U USA
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1966 Grothendieck, Alexander Berlin Germany 38 U of Paris France
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1966 Smale, Stephen Flint MI USA 36 UC Berkeley USA
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1970 Baker, Alan London UK 31 Cambridge U UK
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1970 Hironaka, Heisuke Yamaguchi-ken Japan 39 Harvard U USA
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1970 Novikov, Serge Gorki USSR 32 Moscow U USSR
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1970 Thompson, John Ottawa KA USA 37 U of Chicago USA
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1974 Bombieri, Enrico Milan Italy 33 U of Pisa Italy
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1974 Mumford, David Worth, Sussex UK 37 Harvard U USA
|
|
1978 Deligne, Pierre Brussels Belgium 33 IHES France
|
|
1978 Fefferman, Charles Washington DC USA 29 Princeton U USA
|
|
1978 Margulis, Gregori Moscow USSR 32 InstPrblmInfTrans USSR
|
|
1978 Quillen, Daniel Orange NJ USA 38 MIT USA
|
|
1982 Connes, Alain Draguignan France 35 IHES France
|
|
1982 Thurston, William Washington DC USA 35 Princeton U USA
|
|
1982 Yau, Shing-Tung Kwuntung China 33 IAS USA
|
|
1986 Donaldson, Simon Cambridge UK 27 Oxford U UK
|
|
1986 Faltings, Gerd 1954 Germany 32 Princeton U USA
|
|
1986 Freedman, Michael Los Angeles CA USA 35 UC San Diego USA
|
|
1990 Drinfeld, Vladimir Kharkov USSR 36 Phys.Inst.Kharkov USSR
|
|
1990 Jones, Vaughan Gisbourne N Zealand 38 UC Berkeley USA
|
|
1990 Mori, Shigefumi Nagoya Japan 39 U of Kyoto? Japan
|
|
1990 Witten, Edward Baltimore USA 38 Princeton U/IAS USA
|
|
|
|
References :
|
|
|
|
International Mathematical Congresses, An Illustrated History 1893-1986,
|
|
Revised Edition, Including 1986, by Donald J.Alberts, G. L. Alexanderson
|
|
and Constance Reid, Springer Verlag, 1987.
|
|
|
|
Tropp, Henry S., ``The origins and history of the Fields Medal,''
|
|
Historia Mathematica, 3(1976), 167-181.
|
|
|
|
|
|
8Q: What is 0^0 ?
|
|
|
|
A: According to some Calculus textbooks, 0^0 is an "indeterminate
|
|
form". When evaluating a limit of the form 0^0, then you need
|
|
to know that limits of that form are called "indeterminate forms",
|
|
and that you need to use a special technique such as L'Hopital's
|
|
rule to evaluate them. Otherwise, 0^0=1 seems to be the most
|
|
useful choice for 0^0. This convention allows us to extend
|
|
definitions in different areas of mathematics that otherwise would
|
|
require treating 0 as a special case. Notice that 0^0 is a
|
|
discontinuity of the function x^y.
|
|
|
|
Rotando & Korn show that if f and g are real functions that vanish
|
|
at the origin and are _analytic_ at 0 (infinitely differentiable is
|
|
not sufficient), then f(x)^g(x) approaches 1 as x approaches 0 from
|
|
the right.
|
|
|
|
From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):
|
|
|
|
"Some textbooks leave the quantity 0^0 undefined, because the
|
|
functions x^0 and 0^x have different limiting values when x
|
|
decreases to 0. But this is a mistake. We must define
|
|
|
|
x^0 = 1 for all x,
|
|
|
|
if the binomial theorem is to be valid when x=0, y=0, and/or x=-y.
|
|
The theorem is too important to be arbitrarily restricted! By
|
|
contrast, the function 0^x is quite unimportant."
|
|
|
|
Published by Addison-Wesley, 2nd printing Dec, 1988.
|
|
|
|
References:
|
|
|
|
H. E. Vaughan, The expression '0^0', Mathematics Teacher 63 (1970),
|
|
pp.111-112.
|
|
|
|
Louis M. Rotando & Henry Korn, "The Indeterminate Form 0^0",
|
|
Mathematics Magazine, Vol. 50, No. 1 (January 1977), pp. 41-42.
|
|
|
|
L. J. Paige, A note on indeterminate forms, American Mathematical
|
|
Monthly, 61 (1954), 189-190; reprinted in the Mathematical
|
|
Association of America's 1969 volume, Selected Papers on Calculus,
|
|
pp. 210-211.
|
|
|
|
|
|
|
|
9Q: Why is 0.9999... = 1?
|
|
|
|
A: In modern mathematics, the string of symbols "0.9999..." is
|
|
understood to be a shorthand for "the infinite sum 9/10 + 9/100
|
|
+ 9/1000 + ...." This in turn is shorthand for "the limit of the
|
|
sequence of real numbers 9/10, 9/10 + 9/100, 9/10 + 9/100 + 9/1000,
|
|
..." Using the well-known epsilon-delta definition of limit, one
|
|
can easily show that this limit is 1. The statement that
|
|
0.9999... = 1 is simply an abbreviation of this fact.
|
|
|
|
oo m
|
|
--- 9 --- 9
|
|
0.999... = > ---- = lim > ----
|
|
--- 10^n m->oo --- 10^n
|
|
n=1 n=1
|
|
Choose epsilon > 0. Suppose delta = 1/-log_10 epsilon, thus
|
|
epsilon = 10^(-1/delta). For every m>1/delta we have that
|
|
|
|
| m |
|
|
| --- 9 | 1 1
|
|
| > ---- - 1 | = ---- < ------------ = epsilon
|
|
| --- 10^n | 10^m 10^(1/delta)
|
|
| n=1 |
|
|
|
|
So by the (epsilon-delta) definition of the limit we have
|
|
m
|
|
--- 9
|
|
lim > ---- = 1
|
|
m->oo --- 10^n
|
|
n=1
|
|
|
|
|
|
An *informal* argument could be given by noticing that the following
|
|
sequence of "natural" operations has as a consequence 1 = 0.9999....
|
|
Therefore it's "natural" to assume 1 = 0.9999.....
|
|
|
|
x = 0.99999....
|
|
10x = 9.99999....
|
|
10x - x = 9
|
|
9x = 9
|
|
x = 1
|
|
Thus
|
|
1 = 0.99999....
|
|
|
|
References:
|
|
|
|
E. Hewitt & K. Stromberg, Real and Abstract Analysis,
|
|
Springer-Verlag, Berlin, 1965.
|
|
|
|
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.
|
|
|
|
|
|
|
|
10Q: Where I can get pi up to a few hundred thousand digits of pi?
|
|
Does anyone have an algorithm to compute pi to those zillion
|
|
decimal places?
|
|
|
|
|
|
A: MAPLE or MATHEMATICA can give you 10,000 digits of Pi in a blink,
|
|
and they can compute another 20,000-500,000 overnight (range depends
|
|
on hardware platform).
|
|
|
|
It is possible to retrieve 1.25+ million digits of pi via anonymous
|
|
ftp from the site wuarchive.wustl.edu, in the files pi.doc.Z and
|
|
pi.dat.Z which reside in subdirectory doc/misc/pi.
|
|
|
|
References :
|
|
(This is a short version for a more comprhensive list contact
|
|
Juhana Kouhia at jk87377@cc.tut.fi)
|
|
|
|
J. M. Borwein, P. B. Borwein, and D. H. Bailey, "Ramanujan,
|
|
Modular Equations, and Approximations to Pi", American Mathematical
|
|
Monthly, vol. 96, no. 3 (March 1989), p. 201 - 220.
|
|
|
|
P. Beckman
|
|
A history of pi
|
|
Golem Press, CO, 1971 (fourth edition 1977)
|
|
|
|
J.M. Borwein and P.B. Borwein
|
|
The arithmetic-geometric mean and fast computation of elementary
|
|
functions
|
|
SIAM Review, Vol. 26, 1984, pp. 351-366
|
|
|
|
J.M. Borwein and P.B. Borwein
|
|
More quadratically converging algorithms for pi
|
|
Mathematics of Computation, Vol. 46, 1986, pp. 247-253
|
|
|
|
J.M. Borwein and P.B. Borwein
|
|
Pi and the AGM - a study in analytic number theory and
|
|
computational complexity
|
|
Wiley, New York, 1987
|
|
|
|
Shlomo Breuer and Gideon Zwas
|
|
Mathematical-educational aspects of the computation of pi
|
|
Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2, 1984,
|
|
pp. 231-244
|
|
|
|
Y. Kanada and Y. Tamura
|
|
Calculation of pi to 10,013,395 decimal places based on the
|
|
Gauss-Legendre algorithm and Gauss arctangent relation
|
|
Computer Centre, University of Tokyo, 1983
|
|
|
|
Morris Newman and Daniel Shanks
|
|
On a sequence arising in series for pi
|
|
Mathematics of computation, Vol. 42, No. 165, Jan 1984,
|
|
pp. 199-217
|
|
|
|
E. Salamin
|
|
Computation of pi using arithmetic-geometric mean
|
|
Mathematics of Computation, Vol. 30, 1976, pp. 565-570
|
|
|
|
D. Shanks and J.W. Wrench, Jr.
|
|
Calculation of pi to 100,000 decimals
|
|
Mathematics of Computation, Vol. 16, 1962, pp. 76-99
|
|
|
|
Daniel Shanks
|
|
Dihedral quartic approximations and series for pi
|
|
J. Number Theory, Vol. 14, 1982, pp.397-423
|
|
|
|
David Singmaster
|
|
The legal values of pi
|
|
The Mathematical Intelligencer, Vol. 7, No. 2, 1985
|
|
|
|
Stan Wagon
|
|
Is pi normal?
|
|
The Mathematical Intelligencer, Vol. 7, No. 3, 1985
|
|
|
|
J.W. Wrench, Jr.
|
|
The evolution of extended decimal approximations to pi
|
|
The Mathematics Teacher, Vol. 53, 1960, pp. 644-650
|
|
|
|
|
|
|
|
|
|
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.
|
|
|
|
|
|
|
|
12Q: What is the formula for the "Surface Area" of a sphere in
|
|
Euclidean N-Space. That is, of course, the volume of the N-1
|
|
solid which comprises the boundary of an N-Sphere.
|
|
|
|
A: The volume of a ball is the easiest formula to remember: It's r^N
|
|
times pi^(N/2)/(N/2)!. The only hard part is taking the factorial
|
|
of a half-integer. The real definition is that x! = Gamma(x+1), but
|
|
if you want a formula, it's:
|
|
|
|
(1/2+n)! = sqrt(pi)*(2n+2)!/(n+1)!/4^(n+1)
|
|
|
|
To get the surface area, you just differentiate to get
|
|
N*pi^(N/2)/(N/2)!*r^(N-1).
|
|
|
|
There is a clever way to obtain this formula using Gaussian
|
|
integrals. First, we note that the integral over the line of
|
|
e^(-x^2) is sqrt(pi). Therefore the integral over N-space of
|
|
e^(-x_1^2-x_2^2-...-x_N^2) is sqrt(pi)^n. Now we change to
|
|
spherical coordinates. We get the integral from 0 to infinity
|
|
of V*r^(N-1)*e^(-r^2), where V is the surface volume of a sphere.
|
|
Integrate by parts repeatedly to get the desired formula.
|
|
|
|
13Q: Anyone knows a name (or a closed form) for
|
|
|
|
f(x)^f(x)=x
|
|
|
|
|
|
Solving for f one finds a "continued fraction"-like answer
|
|
|
|
|
|
f(x) = log x
|
|
-----
|
|
log (log x
|
|
------
|
|
...........
|
|
|
|
A: This question has been repeated here from time to time over the
|
|
years, and no one seems to have heard of any published work on it,
|
|
nor a published name for it (D. Merrit proposes "lx" due to its
|
|
(very) faint resemblence to log). It's not an analytic function.
|
|
|
|
The "continued fraction" form for its numeric solution is highly
|
|
unstable in the region of its minimum at 1/e (because the graph is
|
|
quite flat there yet logarithmic approximation oscillates wildly),
|
|
although it converges fairly quickly elsewhere. To compute its value
|
|
near 1/e, I used the bisection method with good results. Bisection
|
|
in other regions converges much more slowly than the "logarithmic
|
|
continued fraction" form, so a hybrid of the two seems suitable.
|
|
Note that it's dual valued for the reals (and many valued complex
|
|
for negative reals).
|
|
|
|
A similar function is a "built-in" function in MAPLE called W(x).
|
|
MAPLE considers a solution in terms of W(x) as a closed form (like
|
|
the erf function). W is defined as W(x)*exp(W(x))=x.
|
|
|
|
If anyone ever runs across something published on the subject,
|
|
please post.
|
|
|
|
|
|
14Q: The existence of a projective plane of order 10 has long been
|
|
an outstanding problem in discrete mathematics and finite geometry.
|
|
|
|
A: More precisely, the question is: is it possible to define 111 sets
|
|
(lines) of 11 points each such that:
|
|
for any pair of points there is precisely one line containing them
|
|
both and for any pair of lines there is only one point common to
|
|
them both.
|
|
Analogous questions with n^2 + n + 1 and n + 1 instead of 111 and 11
|
|
have been positively answered only in case n is a prime power.
|
|
For n=6 it is not possible. The n=10 case has been settled as
|
|
not possible either by Clement Lam. See Am. Math. Monthly,
|
|
recent issue. As the "proof" took several years of computer search
|
|
(the equivalent of 2000 hours on a Cray-1) it can be called the most
|
|
time-intensive computer assisted single proof.
|
|
The final steps were ready in January 1989.
|
|
|
|
|
|
15Q: Is there a formula to determine the day of the week, given
|
|
the month, day and year?
|
|
|
|
A: Here is the standard method.
|
|
|
|
A. Take the last two digits of the year.
|
|
B. Divide by 4, discarding any fraction.
|
|
C. Add the day of the month.
|
|
D. Add the month's key value: JFM AMJ JAS OND
|
|
144 025 036 146
|
|
E. Subtract 1 for January or February of a non-leap year.
|
|
F. For a Gregorian date, add 0 for 1900's, 6 for 2000's, 4 for 1700's, 2
|
|
for 1800's; for other years, add or subtract multiples of 400.
|
|
G. For a Julian date, add 1 for 1700's, and 1 for every additional
|
|
century you go back.
|
|
H. Add the year.
|
|
|
|
Now take the remainder when you divide by 7; 0 is Sunday, the first day
|
|
of the week, 1 is Monday, and so on.
|
|
|
|
Another formula is:
|
|
|
|
W == k + [2.6m - 0.2] - 2C + Y + [Y/4] + [C/4] mod 7
|
|
where [] denotes the integer floor function (round down),
|
|
k is day (1 to 31)
|
|
m is month (1 = March, ..., 10 = December, 11 = Jan, 12 = Feb)
|
|
Treat Jan & Feb as months of the preceding year
|
|
C is century ( 1987 has C = 19)
|
|
Y is year ( 1987 has Y = 87 except Y = 86 for jan & feb)
|
|
W is week day (0 = Sunday, ..., 6 = Saturday)
|
|
|
|
This formula is good for the Gregorian calendar
|
|
(introduced 1582 in parts of Europe, adopted in 1752 in Great Britain
|
|
and its colonies, and on various dates in other countries).
|
|
|
|
It handles century & 400 year corrections, but there is still a
|
|
3 day / 10,000 year error which the Gregorian calendar does not take.
|
|
into account. At some time such a correction will have to be
|
|
done but your software will probably not last that long :-) !
|
|
|
|
|
|
References:
|
|
|
|
Winning Ways by Conway, Guy, Berlekamp is supposed to have it.
|
|
|
|
Martin Gardner in "Mathematical Carnaval".
|
|
|
|
Michael Keith and Tom Craver, "The Ultimate Perpetual Calendar?",
|
|
Journal of Recreational Mathematics, 22:4, pp. 280-282, 1990.
|
|
|
|
K. Rosen, "Elementary Number Theory", p. 156.
|
|
|
|
|
|
|
|
16Q: What is the Axiom of Choice? Why is it important? Why some articles
|
|
say "such and such is provable, if you accept the axiom of choice."?
|
|
What are the arguments for and against the axiom of choice?
|
|
|
|
|
|
A: There are several equivalent formulations:
|
|
|
|
-The Cartesian product of nonempty sets is nonempty, even
|
|
if the product is of an infinite family of sets.
|
|
|
|
-Given any set S of mutually disjoint nonempty sets, there is a set C
|
|
containing a single member from each element of S. C can thus be
|
|
thought of as the result of "choosing" a representative from each
|
|
set in S. Hence the name.
|
|
|
|
>Why is it important?
|
|
|
|
All kinds of important theorems in analysis require it. Tychonoff's
|
|
theorem and the Hahn-Banach theorem are examples. AC is equivalent
|
|
to the thesis that every set can be well-ordered. Zermelo's first
|
|
proof of this in 1904 I believe was the first proof in which AC was
|
|
made explicit. AC is especially handy for doing infinite cardinal
|
|
arithmetic, as without it the most you get is a *partial* ordering
|
|
on the cardinal numbers. It also enables you to prove such
|
|
interesting general facts as that n^2 = n for all infinite cardinal
|
|
numbers.
|
|
|
|
> What are the arguments for and against the axiom of choice?
|
|
|
|
The axiom of choice is independent of the other axioms of set theory
|
|
and can be assumed or not as one chooses.
|
|
|
|
(For) All ordinary mathematics uses it.
|
|
|
|
There are a number of arguments for AC, ranging from a priori to
|
|
pragmatic. The pragmatic argument (Zermelo's original approach) is
|
|
that it allows you to do a lot of interesting mathematics. The more
|
|
conceptual argument derives from the "iterative" conception of set
|
|
according to which sets are "built up" in layers, each layer consisting
|
|
of all possible sets that can be constructed out of elements in the
|
|
previous layers. (The building up is of course metaphorical, and is
|
|
suggested only by the idea of sets in some sense consisting of their
|
|
members; you can't have a set of things without the things it's a set
|
|
of). If then we consider the first layer containing a given set S of
|
|
pairwise disjoint nonempty sets, the argument runs, all the elements
|
|
of all the sets in S must exist at previous levels "below" the level
|
|
of S. But then since each new level contains *all* the sets that can
|
|
be formed from stuff in previous levels, it must be that at least by
|
|
S's level all possible choice sets have already been *formed*. This
|
|
is more in the spirit of Zermelo's later views (c. 1930).
|
|
|
|
(Against) It has some supposedly counterintuitive consequences,
|
|
such as the Banach-Tarski paradox. (See next question)
|
|
|
|
Arguments against AC typically target its nonconstructive character:
|
|
it is a cheat because it conjures up a set without providing any
|
|
sort of *procedure* for its construction--note that no *method* is
|
|
assumed for picking out the members of a choice set. It is thus the
|
|
platonic axiom par excellence, boldly asserting that a given set
|
|
will always exist under certain circumstances in utter disregard of
|
|
our ability to conceive or construct it. The axiom thus can be seen
|
|
as marking a divide between two opposing camps in the philosophy of
|
|
mathematics: those for whom mathematics is essentially tied to our
|
|
conceptual capacities, and hence is something we in some sense
|
|
*create*, and those for whom mathematics is independent of any such
|
|
capacities and hence is something we *discover*. AC is thus of
|
|
philosophical as well as mathematical significance.
|
|
|
|
|
|
It should be noted that some interesting mathematics has come out of an
|
|
incompatible axiom, the Axiom of Determinacy (AD). AD asserts that
|
|
any two-person game without ties has a winning strategy for the first or
|
|
second player. For finite games, this is an easy theorem; for infinite
|
|
games with duration less than \omega and move chosen from a countable set,
|
|
you can prove the existence of a counter-example using AC. Jech's book
|
|
"The Axiom of Choice" has a discussion.
|
|
|
|
An example of such a game goes as follows.
|
|
|
|
Choose in advance a set of infinite sequences of integers; call it A.
|
|
Then I pick an integer, then you do, then I do, and so on forever
|
|
(i.e. length \omega). When we're done, if the sequence of integers
|
|
we've chosen is in A, I win; otherwise you win. AD says that one of
|
|
us must have a winning strategy. Of course the strategy, and which
|
|
of us has it, will depend upon A.
|
|
|
|
|
|
From a philosophical/intuitive/pedagogical standpoint, I think Bertrand
|
|
Russell's shoe/sock analogy has a lot to recommend it. Suppose you have an
|
|
infinite collection of pairs of shoes. You want to form a set with one
|
|
shoe from each pair. AC is not necessary, since you can define the set as
|
|
"the set of all left shoes". (Technically, we're using the axiom of
|
|
replacement, one of the basic axioms of Zermelo-Fraenkel (ZF) set theory.)
|
|
If instead you want to form a set containing one sock from each pair of an
|
|
infinite collection of pairs of socks, you now need AC.
|
|
|
|
|
|
References:
|
|
|
|
Maddy, "Believing the Axioms, I", J. Symb. Logic, v. 53, no. 2, June 1988,
|
|
pp. 490-500, and "Believing the Axioms II" in v.53, no. 3.
|
|
|
|
Gregory H. Moore, Zermelo's Axiom of Choice, New York, Springer-Verlag,
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|
1982.
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|
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H. Rubin and J. E. Rubin, Equivalents of the Axiom of Choice, Amsterdam,
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|
North-Holland, 1963.
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|
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A. Fraenkel, Y. Bar-Hillel, and A. Levy, Foundations of Set Theory,
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Amsterdam, North-Holland, 1984 (2nd edition, 2nd printing), pp. 53-86.
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17Q: Cutting a sphere into pieces of larger volume. Is it possible
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|
to cut a sphere into a finite number of pieces and reassemble
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|
into a solid of twice the volume?
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A: This question has many variants and it is best answered explicitly.
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|
Given two polygons of the same area, is it always possible to
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|
dissect one into a finite number of pieces which can be reassembled
|
|
into a replica of the other?
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|
|
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Dissection theory is extensive. In such questions one needs to
|
|
specify
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|
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(A) what a "piece" is, (polygon? Topological disk? Borel-set?
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|
Lebesgue-measurable set? Arbitrary?)
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|
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(B) how many pieces are permitted (finitely many? countably? uncountably?)
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(C) what motions are allowed in "reassembling" (translations?
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|
rotations? orientation-reversing maps? isometries?
|
|
affine maps? homotheties? arbitrary continuous images? etc.)
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|
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(D) how the pieces are permitted to be glued together. The
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|
simplest notion is that they must be disjoint. If the pieces
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|
are polygons [or any piece with a nice boundary] you can permit
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|
them to be glued along their boundaries, ie the interiors of the
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|
pieces disjoint, and their union is the desired figure.
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|
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Some dissection results
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1) We are permitted to cut into FINITELY MANY polygons, to TRANSLATE
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and ROTATE the pieces, and to glue ALONG BOUNDARIES;
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|
then Yes, any two equal-area polygons are equi-decomposable.
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|
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|
This theorem was proven by Bolyai and Gerwien independently, and has
|
|
undoubtedly been independently rediscovered many times. I would not
|
|
be surprised if the Greeks knew this.
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|
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|
The Hadwiger-Glur theorem implies that any two equal-area polygons are
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|
equi-decomposable using only TRANSLATIONS and ROTATIONS BY 180
|
|
DEGREES.
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|
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|
2) THM (Hadwiger-Glur, 1951) Two equal-area polygons P,Q are
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|
equi-decomposable by TRANSLATIONS only, iff we have equality of these
|
|
two functions: PHI_P() = PHI_Q()
|
|
Here, for each direction v (ie, each vector on the unit circle in the
|
|
plane), let PHI_P(v) be the sum of the lengths of the edges of P which
|
|
are perpendicular to v, where for such an edge, its length is positive
|
|
if v is an outward normal to the edge and is negative if v is an
|
|
inward normal to the edge.
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3) In dimension 3, the famous "Hilbert's third problem" is:
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|
"If P and Q are two polyhedra of equal volume, are they
|
|
equi-decomposable by means of translations and rotations, by
|
|
cutting into finitely many sub-polyhedra, and gluing along
|
|
boundaries?"
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|
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|
The answer is "NO" and was proven by Dehn in 1900, just a few months
|
|
after the problem was posed. (Ueber raumgleiche polyeder, Goettinger
|
|
Nachrichten 1900, 345-354). It was the first of Hilbert's problems
|
|
to be solved. The proof is nontrivial but does *not* use the axiom
|
|
of choice.
|
|
|
|
"Hilbert's Third Problem", by V.G.Boltianskii, Wiley 1978.
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4) Using the axiom of choice on non-countable sets, you can prove
|
|
that a solid sphere can be dissected into a finite number of
|
|
pieces that can be reassembled to two solid spheres, each of
|
|
same volume of the original. No more than nine pieces are needed.
|
|
|
|
This construction is known as the "Banach-Tarski" paradox or the
|
|
"Banach-Tarski-Hausdorff" paradox (Hausdorff did an early version of
|
|
it). The "pieces" here are non-measurable sets, and they are
|
|
assembled *disjointly* (they are not glued together along a boundary,
|
|
unlike the situation in Bolyai's thm.)
|
|
An excellent book on Banach-Tarski is:
|
|
|
|
"The Banach-Tarski Paradox", by Stan Wagon, 1985, Cambridge
|
|
University Press.
|
|
|
|
Also read in the Mathematical Intelligencier an article on
|
|
the Banach-Tarski Paradox.
|
|
|
|
The pieces are not (Lebesgue) measurable, since measure is preserved
|
|
by rigid motion. Since the pieces are non-measurable, they do not
|
|
have reasonable boundaries. For example, it is likely that each piece's
|
|
topological-boundary is the entire ball.
|
|
|
|
The full Banach-Tarski paradox is stronger than just doubling the
|
|
ball. It states:
|
|
|
|
5) Any two bounded subsets (of 3-space) with non-empty interior, are
|
|
equi-decomposable by translations and rotations.
|
|
|
|
This is usually illustrated by observing that a pea can be cut up
|
|
into finitely pieces and reassembled into the Earth.
|
|
|
|
The easiest decomposition "paradox" was observed first by Hausdorff:
|
|
|
|
6) The unit interval can be cut up into COUNTABLY many pieces which,
|
|
by *translation* only, can be reassembled into the interval of
|
|
length 2.
|
|
|
|
This result is, nowadays, trivial, and is the standard example of a
|
|
non-measurable set, taught in a beginning graduate class on measure
|
|
theory.
|
|
|
|
|
|
References:
|
|
|
|
In addition to Wagon's book above, Boltyanskii has written at least
|
|
two works on this subject. An elementary one is:
|
|
|
|
"Equivalent and equidecomposable figures"
|
|
|
|
in Topics in Mathematics published by D.C. HEATH AND CO., Boston. It
|
|
is a translation from the 1956 work in Russian.
|
|
|
|
Also, the article "Scissor Congruence" by Dubins, Hirsch and ?,
|
|
which appeared about 20 years ago in the Math Monthly, has a pretty
|
|
theorem on decomposition by Jordan arcs.
|
|
|
|
|
|
``Banach and Tarski had hoped that the physical absurdity of this
|
|
theorem would encourage mathematicians to discard AC. They were
|
|
dismayed when the response of the math community was `Isn't AC great?
|
|
How else could we get such unintuitive results?' ''
|
|
|
|
|
|
18Q: Is there a theory of quaternionic analytic functions, that is, a four-
|
|
dimensional analog to the theory of complex analytic functions?
|
|
|
|
A. Yes. This was developed in the 1930s by the mathematician
|
|
Fueter. It is based on a generalization of the Cauchy-Riemann
|
|
equations, since the possible alternatives of power series expansions
|
|
or quaternion differentiability do not produce useful theories.
|
|
A number of useful integral theorems follow from the theory.
|
|
Sudbery provides an excellent review. Deavours covers some of the same
|
|
material less thoroughly. Brackx discusses a further generalization
|
|
to arbitrary Clifford algebras.
|
|
|
|
|
|
Anthony Sudbery, Quaternionic Analysis, Proc. Camb. Phil. Soc.,
|
|
vol. 85, pp 199-225, 1979.
|
|
|
|
Cipher A. Deavours, The Quaternion Calculus, Am. Math. Monthly,
|
|
vol. 80, pp 995-1008, 1973.
|
|
|
|
F. Brackx and R. Delanghe and F. Sommen, Clifford analysis,
|
|
Pitman, 1983.
|
|
|
|
|
|
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? [small biography in peparation]
|
|
|
|
|
|
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 occuring 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.
|
|
|
|
|
|
|
|
21Q.- Why is there no Nobel in mathematics? #
|
|
|
|
[answer is forecoming]
|
|
|
|
22Q.- General References and textbooks... #
|
|
|
|
[a list of general references and most commonly used textbooks]
|
|
[ ]
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--------------------------------------------------------------------------
|
|
Questions and Answers _Compiled_ by:
|
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|
|
Alex Lopez-Ortiz alopez-o@maytag.UWaterloo.ca
|
|
Deparment of Computer Science University of Waterloo
|
|
Waterloo, Ontario Canada
|