From: alopez-o@maytag.uwaterloo.ca (Alex Lopez-Ortiz)
Subject: sci.math: Frequently Asked Questions
Followup-To: sci.math
Summary: (version 3.4)
Originator: alopez-o@maytag.uwaterloo.ca
Reply-To: alopez-o@maytag.uwaterloo.ca
Organization: University of Waterloo
Date: Tue, 6 Oct 1992 13:05:12 GMT


Archive-Name: sci-math-faq

Version: $Id: sci-math-faq,v 3.5 92/08/11 13:04:00 $

This is a list of frequently asked questions for sci.math (version 3.5).
Any contributions/suggestions/corrections are most welcome. (Thanks to all
the people who already contributed).  Please use * e-mail * on any comment 
concerning the FAQ list.

Changes of version will be important enough to deserve reading the FAQ list
again. Additions are marked with a # on the table of contents. Still you may
kill all versions of FAQ using the * wildcard. (Ask your local unix guru for
ways to do so). The FAQ is available via ftp in rtfm.mit.edu (18.172.1.27).


             Table of Contents
             -----------------

 1Q.- Fermat's Last Theorem, status of ..
 2Q.- Four Colour Theorem, proof of ..
 3Q.- Values of Record Numbers      
 4Q.- General Netiquette
 5Q.- Computer Algebra Systems, application of ..
 6Q.- Computer Algebra Systems, references to ..
 7Q.- Fields Medal, general info ..
 8Q.- 0^0=1. A comprehensive approach ..   
 9Q.- 0.999... = 1. Properties of the real numbers ..
10Q.- Digits of Pi, computation and references ..
11Q.- There are three doors, The Monty Hall problem ..
12Q.- Surface and Volume of the n-ball  
13Q.- f(x)^f(x)=x, name of the function ..
14Q.- Projective plane of order 10 ..   
15Q.- How to compute day of week of a given date .... 
16Q.- Axiom of Choice and/or Continuum Hypothesis? 
17Q.- Cutting a sphere into pieces of larger volume  
18Q.- Pointers to Quaternions


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 rather is intended 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 2,000,000 in
    the general case. The method used was 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.


    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 tha
    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 imples 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 gemoetry. [about which I know epsilon]. 

    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.




2Q:  Has the Four Colour Theorem been solved?
    (Every planar map with regions of simple borders can be coloured 
    with 4 colours in such a way that no two regions sharing
    a non-zero length border have the same colour.)

A:  This theorem was proved with the aid of a computer in 1976.
    The proof shows that if aprox. 1,936  basic forms of maps
    can be coloured with four colours, then any given map can be
    coloured with four colours. A computer progam coloured this 
    basic forms. So far nobody has been able to prove it without 
    using a computer. In principle it is possible to emulate the 
    computer proof by hand computations.

    References:

    K. Appel and W. Haken, Every planar map is four colourable,
    Bulletin of the American Mathematical Society, vol. 82, 1976 pp.711-712.

    K. Appel and W. Haken, Every planar map is four colourable,
    Illinois Journal of Mathematics, vol. 21, 1977, pp. 429-567.

    T. Saaty and Paul Kainen, The Four Colour Theorem: Assault and Conquest,
    McGraw-Hill, 1977. Reprinted by Dover Publications 1986. 

    K. Appel and W. Haken, Every Planar Map is Four Colorable,
    Contemporary Mathematics, vol. 98, American Mathematical Society,
    1989, pp.741.

    F. Bernhart, Math Reviews. 91m:05007, Dec. 1991. (Review of Appel
    and Haken's book).




3Q:  What are the values of:

largest known Mersenne prime?

A:  It is 2^756839 - 1. It was discovered by a Cray-2 in England in 1992.
    It has 227,832 digits.

	
largest known prime?

A:  The largest known prime was 391581*2^216193 - 1.  See Brown, Noll,
    Parady, Smith, Smith, and Zarantonello, Letter to the editor,
    American Mathematical Monthly, vol. 97, 1990, p. 214.

    Now the largest known prime is the Mersenne prime described above.

	
largest known twin primes?
	
A:  The largest known twin primes are 1706595*2^11235 +- 1.
    See B. K. Parady and J. F. Smith and S. E. Zarantonello,
    Smith, Noll and Brown.
    Largest known twin primes, Mathematics of Computation,
    vol.55, 1990, pp. 381-382. 


largest Fermat number with known factorization?

A:  F_11 = (2^(2^11)) + 1 which was  factored by Brent & Morain in
    1988. F9 = (2^(2^9)) + 1 = 2^512 + 1 was factored by 
    A.K. Lenstra, H.W. Lenstra Jr., M.S. Manasse & J.M. Pollard
    in 1990. The factorization for F10 is NOT known.


Are there good algorithms to factor a given integer?

A:  There are several that have subexponential estimated 
    running time, to mention just a few:

        Continued fraction algorithm,
        Class group method,
        Quadratic sieve algorithm,
        Elliptic curve algorithm,
        Number field sieve,
        Dixon's random squares algorithm,
        Valle's two-thirds algorithm,
        Seysen's class group algorithm,

    A.K. Lenstra, H.W. Lenstra Jr., "Algorithms in Number Theory",
    in: J. van Leeuwen (ed.), Handbook of Theoretical Computer 
    Science, Volume A: Algorithms and Complexity, Elsevier, pp. 
    673-715, 1990.


List of record numbers?

A:  Chris Caldwell maintains "THE LARGEST KNOWN PRIMES (ALL KNOWN
    PRIMES WITH 2000 OR MORE DIGITS)"-list. Send him, bf04@UTMartn.bitnet
    (preferred) or kvax@utkvx.UTK.edu, any new gigantic primes (greater
    than 10,000 digits), titanic primes (greater than 1000 digits).


What is the current status on Mersenne primes?

A:  Mersenne primes are primes of the form 2^p-1. For 2^p-1 to be prime we
    must have that p is prime. The following Mersenne primes are known.

    nr            p                                 year  by
    -----------------------------------------------------------------
     1-5   2,3,5,7,13                    in or before the middle ages
     6-7       17,19                     1588  Cataldi
     8          31                       1750  Euler
     9          61                       1883  Pervouchine
    10          89                       1911  Powers
    11          107                      1914  Powers
    12          127                      1876  Lucas
    13-14       521,607                  1952  Robinson
    15-17       1279,2203,2281           1952  Lehmer
    18          3217                     1957  Riesel
    19-20       4253,4423                1961  Hurwitz & Selfridge
    21-23       9689,9941,11213          1963  Gillies
    24          19937                    1971  Tuckerman
    25          21701                    1978  Noll & Nickel
    26          23209                    1979  Noll
    27          44497                    1979  Slowinski & Nelson
    28          86243                    1982  Slowinski
    29          110503                   1988  Colquitt & Welsh jr.
    30          132049                   1983  Slowinski
    31          216091                   1985  Slowinski
    32?         756839                   1992  Slowinski & Gage

   The way to determine if 2^p-1 is prime is to use the Lucas-Lehmer test:
      Lucas_Lehmer_Test(p):
         u := 4
         for i from 3 to p do
            u := u^2-2 mod 2^p-1
         od
         if u == 0 then
            2^p-1 is prime
         else
            2^p-1 is composite
         fi

   The following ranges have been checked completely:
    2 - 355K and  430K - 520K

   More on Mersenne primes and the Lucas-Lehmer test can be found in:
      G.H. Hardy, E.M. Wright, An introduction to the theory of numbers,
      fifth edition, 1979, pp. 16, 223-225.


(Please send updates to alopez-o@maytag.UWaterloo.ca)




4Q:  I think I proved <insert big conjecture>.    OR
    I think I have a bright new idea.

    What should I do?

A:  Are you an expert in the area? If not, please ask first local
    gurus for pointers to related work (the "distribution" field
    may serve well for this purposes). If after reading them you still
    think your *proof is correct*/*idea is new* then send it to the net.


5Q:  I have this complicated symbolic problem (most likely
    a symbolic integral or a DE system) that I can't solve.
    What should I do?

A:  Find a friend with access to a computer algebra system
    like MAPLE, MACSYMA or MATHEMATICA and ask her/him to solve it.
    If packages cannot solve it, then (and only then) ask the net. 


6Q:  Where can I get <Symbolic Computation Package>?
    This is not a comprehensive list. There are other
    Computer Algebra packages available that may better
    suit your needs.

A: Maple 
        Purpose: Symbolic and numeric computation, mathematical
        programming, and mathematical visualization. 
        Contact: Waterloo Maple Software,
        160 Columbia Street West,
        Waterloo, Ontario, Canada     N2L 3L3
        Phone: (519) 747-2373 
        wmsi@daisy.uwaterloo.ca wmsi@daisy.waterloo.edu

A: DOE-Macsyma  
        Purpose: Symbolic and mathematical manipulations.
        Contact: National Energy Software Center
        Argonne National Laboratory 9700 South Cass Avenue
        Argonne, Illinois 60439 
        Phone: (708) 972-7250

A: Pari    
        Purpose: Number-theoretic computations and simple numerical
        analysis.
        Available for Sun 3, Sun 4, generic 32-bit Unix, and
        Macintosh II. This is a free package, available by ftp from
        math.ucla.edu (128.97.64.16).
        Contact: questions about pari can be sent to pari@mizar.greco-prog.fr

A: Mathematica
        Purpose: Mathematical computation and visualization,
        symbolic programming. 
        Contact: Wolfram Research, Inc. 
        100 Trade Center Drive Champaign,
        IL 61820-7237
        Phone: 1-800-441-MATH

A: Macsyma
        Purpose: Symbolic and mathematical manipulations.
	Contact: Symbolics, Inc.
	8 New England Executive Park East
	Burlington, Massachusetts 01803
	United States of America
	(617) 221-1250
	macsyma@Symbolics.COM

A: Matlab
        Purpose: `matrix laboratory' for tasks involving 
	matrices, graphics and general numerical computation.
	Contact: The MathWorks, Inc.
     	21 Eliot Street
     	South Natick, MA 01760
     	508-653-1415
     	info@mathworks.com

A: Cayley
        Purpose: Computation in algebraic and combinatorial structures
        such as groups, rings, fields, modules and graphs.
        Available for: SUN 3, SUN 4, IBM running AIX or VM, DEC VMS, others
        Contact: Computational Algebra Group
        University of Sydney
        NSW 2006
        Australia
        Phone:  (61) (02) 692 3338
        Fax: (61) (02) 692 4534
        cayley@maths.su.oz.au



7Q:  Let P be a property about the Fields Medal. Is P(x) true?

A:  There are a few gaps in the list. If you know any of the
    missing information (or if you notice any mistakes), 
    please send me e-mail.

Year Name               Birthplace              Age Institution
---- ----               ----------              --- -----------
1936 Ahlfors, Lars      Helsinki       Finland   29 Harvard U         USA
1936 Douglas, Jesse     New York NY    USA       39 MIT               USA
1950 Schwartz, Laurent  Paris          France    35 U of Nancy        France
1950 Selberg, Atle      Langesund      Norway    33 Adv.Std.Princeton USA 
1954 Kodaira, Kunihiko  Tokyo          Japan     39 Princeton U       USA
1954 Serre, Jean-Pierre Bages          France    27 College de France France
1958 Roth, Klaus        Breslau        Germany   32 U of London       UK
1958 Thom, Rene         Montbeliard    France    35 U of Strasbourg   France
1962 Hormander, Lars    Mjallby        Sweden    31 U of Stockholm    Sweden
1962 Milnor, John       Orange NJ      USA       31 Princeton U       USA
1966 Atiyah, Michael    London         UK        37 Oxford U          UK
1966 Cohen, Paul        Long Branch NJ USA       32 Stanford U        USA
1966 Grothendieck, Alexander Berlin    Germany   38 U of Paris        France
1966 Smale, Stephen     Flint MI       USA       36 UC Berkeley       USA
1970 Baker, Alan        London         UK        31 Cambridge U       UK
1970 Hironaka, Heisuke  Yamaguchi-ken  Japan     39 Harvard U         USA
1970 Novikov, Serge     Gorki          USSR      32 Moscow U          USSR
1970 Thompson, John     Ottawa KA      USA       37 U of Chicago      USA
1974 Bombieri, Enrico   Milan          Italy     33 U of Pisa         Italy
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     Auckland       N Zealand 38 UC Berkeley       USA
1990 Mori, Shigefumi    Nagoya         Japan     39 U of Kyoto?       Japan
1990 Witten, Edward     ?              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.

    Another reference is:
 
    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 :

    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...

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

    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 netlib@peregrine.com an email
    message consisting of the text

               send switch


    References
    
    American Mathematical Monthly, January 1992.


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,
    1982.

    H. Rubin and J. E. Rubin, Equivalents of the Axiom of Choice, Amsterdam,
     North-Holland, 1963.

    A. Fraenkel, Y.  Bar-Hillel, and A. Levy, Foundations of Set Theory, 
    Amsterdam, North-Holland, 1984 (2nd edition, 2nd printing), pp. 53-86.



17Q:  Cutting a sphere into pieces of larger volume. Is it possible
    to cut a sphere into a finite number of pieces and reassemble 
    into a solid of twice the volume?

A:  This question has many variants and it is best answered explicitly.
    Given two polygons of the same area, is it always possible to
    dissect one into a finite number of pieces which can be reassembled
    into a replica of the other?

    Dissection theory is extensive.  In such questions one needs to
    specify

     (A) what a "piece" is,  (polygon?  Topological disk?  Borel-set? 
         Lebesgue-measurable set?  Arbitrary?)

     (B) how many pieces are permitted (finitely many? countably? uncountably?)

     (C) what motions are allowed in "reassembling" (translations?
         rotations?  orientation-reversing maps?  isometries?  
         affine maps?  homotheties?  arbitrary continuous images?  etc.)

     (D) how the pieces are permitted to be glued together.  The
         simplest notion is that they must be disjoint.  If the pieces
         are polygons [or any piece with a nice boundary] you can permit
         them to be glued along their boundaries, ie the interiors of the
         pieces disjoint, and their union is the desired figure.


    Some dissection results

     1) We are permitted to cut into FINITELY MANY polygons, to TRANSLATE
        and ROTATE the pieces, and to glue ALONG BOUNDARIES;
        then Yes, any two equal-area polygons are equi-decomposable.

        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.

        The Hadwiger-Glur theorem implies that any two equal-area polygons are
        equi-decomposable using only TRANSLATIONS and ROTATIONS BY 180
        DEGREES. 

     2) THM (Hadwiger-Glur, 1951) Two equal-area polygons P,Q are
        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.


     3) In dimension 3, the famous "Hilbert's third problem" is:
     
       "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?" 

        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.


     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 equidecompsable 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.





--------------------------------------------------------------------------
Questions and Answers _Compiled_ by:

Alex Lopez-Ortiz                              alopez-o@maytag.UWaterloo.ca
Deparment of Computer Science                       University of Waterloo
Waterloo, Ontario                                                   Canada
Newsgroups: sci.math