Newsgroups: sci.math,news.answers
From: alopez-o@maytag.uwaterloo.ca (Alex Lopez-Ortiz)
Subject: sci.math: Frequently Asked Questions
Message-ID:
Summary: (version 3.8)
Organization: University of Waterloo
Date: Thu, 21 Jan 1993 14:05:10 GMT
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Archive-Name: sci-math-faq
Version: $Id: sci-math-faq,v 3.8 92/12/26 18:45:00 $
This is a list of Frequently Asked Questions for sci.math (version 3.8).
Any contributions/suggestions/corrections are most welcome. Please use
* e-mail * on any comment concerning the FAQ list.
Changes and additions are marked with a # on the table of contents.
This FAQ list (and most others, for that matter) is available via anonymous
ftp at rtfm.mit.edu (18.172.1.27).
The list of contributors to this FAQ list is too large to include here;
but thanks are due to all of them (you know who you are folks).
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, Master Mind and
other games .. #
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
19Q.- Erdos Number #
20Q.- Odd Perfect Number #
21Q.- Why is there no Nobel in mathematics? #
22Q.- General References and textbooks... #
1Q: What is the current status of Fermat's last theorem?
(There are no positive integers x,y,z, and n > 2 such that
x^n + y^n = z^n)
I heard that claimed to have proved it but later
on the proof was found to be wrong. ...
(wlog we assume x,y,z to be relatively prime)
A: The status of FLT has remained remarkably constant. Every few
years, someone claims to have a proof ... but oh, wait, not quite.
Meanwhile, it is proved true for ever greater values of the exponent
(but not all of them), and ties are shown between it and other
conjectures (if only we could prove one of them), and ... so it has
been for quite some time. It has been proved that for each
exponent, there are at most a finite number of counter-examples to
FLT.
Here is a brief survey of the status of FLT. It is not intended to
be 'deep', but it is rather for non-specialists.
The theorem is broken into 2 cases. The first case assumes
(abc,n) = 1. The second case is the general case.
What has been PROVED
--------------------
First Case.
It has been proven true up to 7.568x10^17 by the work of Wagstaff &
Tanner, Granville&Monagan, and Coppersmith. They all used extensions
of the Wiefrich criteria and improved upon work performed by
Gunderson and Shanks&Williams.
The first case has been proven to be true for an infinite number of
exponents by Adelman, Frey, et. al. using a generalization of the
Sophie Germain criterion
Second Case:
It has been proven true up to n = 150,000 by Tanner & Wagstaff. The
work used new techniques for computing Bernoulli numbers mod p and
improved upon work of Vandiver. The work involved computing the
irregular primes up to 150,000. FLT is true for all regular primes
by a theorem of Kummer. In the case of irregular primes, some
additional computations are needed.
UPDATE :
Fermat's Last Theorem has been proved true up to exponent 4,000,000
in the general case. The method used was essentially that of Wagstaff:
enumerating and eliminating irregular primes by Bernoulli number
computations. The computations were performed on a set of NeXT
computers by Richard Crandall et al.
Since the genus of the curve a^n + b^n = 1, is greater than or equal
to 2 for n > 3, it follows from Mordell's theorem [proved by
Faltings], that for any given n, there are at most a finite number
of solutions.
Conjectures
-----------
There are many open conjectures that imply FLT. These conjectures
come from different directions, but can be basically broken into
several classes: (and there are interrelationships between the
classes)
(a) conjectures arising from Diophantine approximation theory such
as the ABC conjecture, the Szpiro conjecture, the Hall conjecture,
etc.
For an excellent survey article on these subjects see the article
by Serge Lang in the Bulletin of the AMS, July 1990 entitled
"Old and new conjectured diophantine inequalities".
Masser and Osterle formulated the following known as the ABC
conjecture:
Given epsilon > 0, there exists a number C(epsilon) such that for
any set of non-zero, relatively prime integers a,b,c such that
a+b = c we have
max( |a|, |b|, |c|) <= C(epsilon) N(abc)^(1 + epsilon)
where N(x) is the product of the distinct primes dividing x.
It is easy to see that it implies FLT asymptotically. The conjecture
was motivated by a theorem, due to Mason that essentially says the
ABC conjecture IS true for polynomials.
The ABC conjecture also implies Szpiro's conjecture [and vice-versa]
and Hall's conjecture. These results are all generally believed to
be true.
There is a generalization of the ABC conjecture [by Vojta] which is
too technical to discuss but involves heights of points on
non-singular algebraic varieties . Vojta's conjecture also implies
Mordell's theorem [already known to be true]. There are also a
number of inter-twined conjectures involving heights on elliptic
curves that are related to much of this stuff. For a more complete
discussion, see Lang's article.
(b) conjectures arising from the study of elliptic curves and
modular forms. -- The Taniyama-Weil-Shmimura conjecture.
There is a very important and well known conjecture known as the
Taniyama-Weil-Shimura conjecture that concerns elliptic curves.
This conjecture has been shown by the work of Frey, Serre, Ribet,
et. al. to imply FLT uniformly, not just asymptotically as with the
ABC conj.
The conjecture basically states that all elliptic curves can be
parameterized in terms of modular forms.
There is new work on the arithmetic of elliptic curves. Sha, the
Tate-Shafarevich group on elliptic curves of rank 0 or 1. By the way
an interesting aspect of this work is that there is a close
connection between Sha, and some of the classical work on FLT. For
example, there is a classical proof that uses infinite descent to
prove FLT for n = 4. It can be shown that there is an elliptic curve
associated with FLT and that for n=4, Sha is trivial. It can also be
shown that in the cases where Sha is non-trivial, that
infinite-descent arguments do not work; that in some sense 'Sha
blocks the descent'. Somewhat more technically, Sha is an
obstruction to the local-global principle [e.g. the Hasse-Minkowski
theorem].
(c) Conjectures arising from some conjectured inequalities involving
Chern classes and some other deep results/conjectures in arithmetic
algebraic geometry.
I can't describe these results since I don't know the math. Contact
Barry Mazur [or Serre, or Faltings, or Ribet, or ...]. Actually the
set of people who DO understand this stuff is fairly small.
The diophantine and elliptic curve conjectures all involve deep
properties of integers. Until these conjecture were tied to FLT,
FLT had been regarded by most mathematicians as an isolated problem;
a curiosity. Now it can be seen that it follows from some deep and
fundamental properties of the integers. [not yet proven but
generally believed].
This synopsis is quite brief. A full survey would run to many pages.
References:
[1] J.P.Butler, R.E.Crandall, & R.W.Sompolski
"Irregular Primes to One Million"
Math. Comp. 59 (October 1992) pp. 717-722
H.M. Edwards, Fermat's Last Theorem, A Genetic Introduction to
Algebraic Number Theory, Springer Verlag, New York, 1977
P. Ribenboim, Thirteen Lectures on Fermat's Last Theorem,
Springer Verlag, New York, 1979
Number Theory Related to Fermat's Last Theorem, Neal Koblitz, editor,
Birkh\"auser Boston, Inc., 1982, ISBN 3-7643-3104-6
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 program 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 is the Mersenne prime described above.
The previous record holder, and the largest known non-Mersenne prime,
is 391581*2^216193-1. See Brown, Noll, Parady, Smith, Smith, and
Zarantonello, Letter to the editor, American Mathematical Monthly,
vol. 97, 1990, p. 214. Throughout history, the largest known prime
has almost always been a Mersenne prime; the period between Brown
et al's discovery in Aug 1989 and Slowinski & Gage's in March 1992
is one of the few exceptions.
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 mail to
bf04@UTMartn.bitnet (preferred) or kvax@utkvx.UTK.edu, on 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 . 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 ?
This is not a comprehensive list. There are other Computer Algebra
packages available that may better suit your needs. There is also
a FAQ list in the group sci.math.symbolics. It includes a much larger
list of vendors and developers. (The FAQ list can be obtained from
rtfm.mit.edu via anonymous ftp).
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 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,
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 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:
Alex Lopez-Ortiz alopez-o@maytag.UWaterloo.ca
Deparment of Computer Science University of Waterloo
Waterloo, Ontario Canada