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Archive-Name: sci-math-faq/part1
Last-modified: October 26, 1993
Version: 5.0


This is the list of Frequently Asked Questions for sci.math (version 5.0).
Any contributions/suggestions/corrections are most welcome. Please use
* e-mail * on any comment concerning the FAQ list.


Section 1 of 3, questions 1Q to 5Q.


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

 1Q.- Fermat's Last Theorem, status of ..
 2Q.- Values of Record Numbers
 3Q.- Formula for prime numbers...
 4Q.- Digits of Pi, computation and references
 5Q.- Odd Perfect Number
 6Q.- Computer Algebra Systems, application of ..
 7Q.- Computer Algebra Systems, references to ..
 8Q.- Fields Medal, general info ..
 9Q.- Four Colour Theorem, proof of ..
10Q.- 0^0=1. A comprehensive approach
11Q.- 0.999... = 1. Properties of the real numbers ..
12Q.- There are three doors, The Monty Hall problem, Master Mind and
      other games ..
13Q.- Surface and Volume of the n-ball
14Q.- f(x)^f(x)=x, name of the function ..
15Q.- Projective plane of order 10 ..
16Q.- How to compute day of week of a given date
17Q.- Axiom of Choice and/or Continuum Hypothesis?
18Q.- Cutting a sphere into pieces of larger volume
19Q.- Pointers to Quaternions
20Q.- Erdos Number
21Q.- Why is there no Nobel in mathematics?
22Q.- General References and textbooks...
23Q.- Interest Rate...
24Q.- Euler's formula e^(i Pi) = - 1 ...


1Q:  What is the current status of Fermat's last theorem?


and

    Did Fermat prove this theorem?


    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. ...

A:  The status of FLT has remained remarkably constant.  Every few
    years, someone claims to have a proof ... but oh, wait, not quite.

    UPDATE... UPDATE... UPDATE

    Andrew Wiles, a researcher at Princeton, Cambridge claims to have
    found a proof.

    The proof was presented in Cambridge, UK during a three day seminar
    to an audience including some of the leading experts in the field.

    The proof is long and cumbersome, but here are some of the first
    few details:

    *From Ken Ribet:

    Here is a brief summary of what Wiles said in his three lectures.

    The method of Wiles borrows results and techniques from lots and lots
    of people.  To mention a few: Mazur, Hida, Flach, Kolyvagin, yours
    truly, Wiles himself (older papers by Wiles), Rubin...  The way he does
    it is roughly as follows.  Start with a mod p representation of the
    Galois group of Q which is known to be modular.  You want to prove that
    all its lifts with a certain property are modular.  This means that the
    canonical map from Mazur's universal deformation ring to its "maximal
    Hecke algebra" quotient is an isomorphism.  To prove a map like this is
    an isomorphism, you can give some sufficient conditions based on
    commutative algebra.  Most notably, you have to bound the order of a
    cohomology group which looks like a Selmer group for Sym^2 of the
    representation attached to a modular form.  The techniques for doing
    this come from Flach; you also have to use Euler systems a la
    Kolyvagin, except in some new geometric guise.

    If you take an elliptic curve over Q, you can look at the
    representation of Gal on the 3-division points of the curve.  If you're
    lucky, this will be known to be modular, because of results of Jerry
    Tunnell (on base change).  Thus, if you're lucky, the problem I
    described above can be solved (there are most definitely some
    hypotheses to check), and then the curve is modular.  Basically, being
    lucky means that the image of the representation of Galois on
    3-division points is GL(2,Z/3Z).

    Suppose that you are unlucky, i.e., that your curve E has a rational
    subgroup of order 3.  Basically by inspection, you can prove that if it
    has a rational subgroup of order 5 as well, then it can't be
    semistable.  (You look at the four non-cuspidal rational points of
    X_0(15).)  So you can assume that E[5] is "nice." Then the idea is to
    find an E' with the same 5-division structure, for which E'[3] is
    modular.  (Then E' is modular, so E'[5] = E[5] is modular.)  You
    consider the modular curve X which parameterizes elliptic curves whose
    5-division points look like E[5].  This is a "twist" of X(5).  It's
    therefore of genus 0, and it has a rational point (namely, E), so it's
    a projective line.  Over that you look at the irreducible covering
    which corresponds to some desired 3-division structure.  You use
    Hilbert irreducibility and the Cebotarev density theorem (in some way
    that hasn't yet sunk in) to produce a non-cuspidal rational point of X
    over which the covering remains irreducible.  You take E' to be the
    curve corresponding to this chosen rational point of X.


    *From the previous version of the FAQ:

    (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].

    *From Karl Rubin:

    Theorem.  If E is a semistable elliptic curve defined over Q,
      then E is modular.

    It has been known for some time, by work of Frey and Ribet, that
    Fermat follows from this.  If u^q + v^q + w^q = 0, then Frey had
    the idea of looking at the (semistable) elliptic curve
    y^2 = x(x-a^q)(x+b^q).  If this elliptic curve comes from a modular
    form, then the work of Ribet on Serre's conjecture shows that there
    would have to exist a modular form of weight 2 on Gamma_0(2).  But
    there are no such forms.

    To prove the Theorem, start with an elliptic curve E, a prime p and let

         rho_p : Gal(Q^bar/Q) -> GL_2(Z/pZ)

    be the representation giving the action of Galois on the p-torsion
    E[p].  We wish to show that a _certain_ lift of this representation
    to GL_2(Z_p) (namely, the p-adic representation on the Tate module
    T_p(E)) is attached to a modular form.  We will do this by using
    Mazur's theory of deformations, to show that _every_ lifting which
    'looks modular' in a certain precise sense is attached to a modular form.

    Fix certain 'lifting data', such as the allowed ramification,
    specified local behavior at p, etc. for the lift. This defines a
    lifting problem, and Mazur proves that there is a universal
    lift, i.e. a local ring R and a representation into GL_2(R) such
    that every lift of the appropriate type factors through this one.

    Now suppose that rho_p is modular, i.e. there is _some_ lift
    of rho_p which is attached to a modular form.  Then there is
    also a hecke ring T, which is the maximal quotient of R with the
    property that all _modular_ lifts factor through T.  It is a
    conjecture of Mazur that R = T, and it would follow from this
    that _every_ lift of rho_p which 'looks modular' (in particular the
    one we are interested in) is attached to a modular form.

    Thus we need to know 2 things:

      (a)  rho_p is modular
      (b)  R = T.

    It was proved by Tunnell that rho_3 is modular for every elliptic
    curve.  This is because PGL_2(Z/3Z) = S_4.  So (a) will be satisfied
    if we take p=3.  This is crucial.

    Wiles uses (a) to prove (b) under some restrictions on rho_p.  Using
    (a) and some commutative algebra (using the fact that T is Gorenstein,
    'basically due to Mazur')  Wiles reduces the statement T = R to
    checking an inequality between the sizes of 2 groups.  One of these
    is related to the Selmer group of the symmetric square of the given
    modular lifting of rho_p, and the other is related (by work of Hida)
    to an L-value.  The required inequality, which everyone presumes is
    an instance of the Bloch-Kato conjecture, is what Wiles needs to verify.

    He does this using a Kolyvagin-type Euler system argument.  This is
    the most technically difficult part of the proof, and is responsible
    for most of the length of the manuscript.  He uses modular
    units to construct what he calls a 'geometric Euler system' of
    cohomology classes.  The inspiration for his construction comes
    from work of Flach, who came up with what is essentially the
    'bottom level' of this Euler system.  But Wiles needed to go much
    farther than Flach did.  In the end, _under_certain_hypotheses_ on rho_p
    he gets a workable Euler system and proves the desired inequality.
    Among other things, it is necessary that rho_p is irreducible.

    Suppose now that E is semistable.

    Case 1.  rho_3 is irreducible.
    Take p=3.  By Tunnell's theorem (a) above is true.  Under these
    hypotheses the argument above works for rho_3, so we conclude
    that E is modular.

    Case 2.  rho_3 is reducible.
    Take p=5.  In this case rho_5 must be irreducible, or else E
    would correspond to a rational point on X_0(15).  But X_0(15)
    has only 4 noncuspidal rational points, and these correspond to
    non-semistable curves.  _If_ we knew that rho_5 were modular,
    then the computation above would apply and E would be modular.

    We will find a new semistable elliptic curve E' such that
    rho_{E,5} = rho_{E',5} and rho_{E',3} is irreducible.  Then
    by Case I, E' is modular.  Therefore rho_{E,5} = rho_{E',5}
    does have a modular lifting and we will be done.

    We need to construct such an E'.  Let X denote the modular
    curve whose points correspond to pairs (A, C) where A is an
    elliptic curve and C is a subgroup of A isomorphic to the group
    scheme E[5].  (All such curves will have mod-5 representation
    equal to rho_E.)  This X is genus 0, and has one rational point
    corresponding to E, so it has infinitely many.  Now Wiles uses a
    Hilbert Irreducibility argument to show that not all rational
    points can be images of rational points on modular curves
    covering X, corresponding to degenerate level 3 structure
    (i.e. im(rho_3) not GL_2(Z/3)).  In other words, an E' of the
    type we need exists.  (To make sure E' is semistable, choose
    it 5-adically close to E.  Then it is semistable at 5, and at
    other primes because rho_{E',5} = rho_{E,5}.)



2Q:  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 1691232 * 1001 * 10^4020 +- 1,
     which is a number with 4030 digits, found by H. Dubner.

    The second largest known twin primes are 4650828 * 1001 * 10^3429 +- 1.
    They were found by H. Dubner

    References:

    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
    [e-mail] (preferred) or [e-mail], 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.



3Q.-  Formula for prime numbers...


     Is there a polynomial which gives all the prime numbers?

      No, there is not. This is a simple exercise to prove.

     Is there a non-constant polynomial that only takes on prime values?

      It has been proved that no such polynomial exists.

      The proof is simple enough that an high school student could probably
      discover it.  See, for example, Ribenboim's book _The Book of Prime
      Number Records_.

     Note, however, by the work of Jones, Sato, Wada, and Wiens, there *is* a
     polynomial in 26 variables such that the set of primes coincides with
     the set of *positive* values taken by this polynomial.  See Ribenboim,
     pp. 147-150.

     But most people would object to the term "formula" restricted to mean
     polynomial.  Can we not use summation signs, factorial, and the floor
     function in our "formula"?  If so, then indeed, there *are* formulas
     for the prime numbers.  Some of them are listed below.

     If we can't, then exactly what operations do you allow and why?

     Indeed, as I have previously argued, a reasonable interpretation of
     the word "formula" is simply "Turing machine that halts on all inputs".
     Under this interpretation, there certainly are halting Turing machines
     which compute the n'th prime number.  However, nobody knows how to
     compute the n'th prime in time polynomial in log n.  That's still
     an open question.

     Herb Wilf has addressed the question, "What is a formula?" in his
     delightful article, "What is an answer?" which appeared in the
     American Mathematical Monthly, 89 (1982), 289-292.  He draws the
     distinction between "formula" and "good formula".  Anyone who claims
     "there is no formula for the prime numbers" should read this article.

     Here are just a few articles that discuss "formulas" for primes.  Almost
     all of these do *not* require computation of the primes "ahead of time".
     Most of them rely on standard mathematical functions such as summation,
     factorial, greatest integer function, etc.


       C. Isenkrahe, Math. Annalen  53 (1900), 42-44.

       W. H. Mills, Bull. Amer. Math. Soc. 53 (1947), 604.

       L. Moser, Math. Mag. 23 (1950), 163-164.

       E. M. Wright, Amer. Math. Monthly 58 (1951), 616-618.  (Correction,
      59 (1952), 99.)

       E. M. Wright, J. Lond. Math. Soc. 29 (1954), 63-71.

       B. R. Srinivasan, J. Indian Math. Soc. 25 (1961), 33-39.

       C. P. Willans, Math. Gazette 48 (1964), 413-415.

       V. C. Harris, Nordisk Mat. Tidskr. 17 (1969), 82.

       U. Dudley, Amer. Math. Monthly 76 (1969), 23-28.

       C. Vanden Eynden, Amer. Math. Monthly 79 (1972), 625.

       S. W. Golomb, Amer. Math. Monthly 81 (1974), 752-754.


      For more references see

       J.O. Shallit, E. Bach, _Algorithmic Number Theory_ (to be published,
       MIT Press).


4Q:  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 20,000-1,000,000 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.

    New York's Chudnovsky brothers have computed 2 billion digits of pi
    on a homebrew computer.

    How is pi calculated to many decimals ?

    There are essentially 3 different methods.

     1) One of the oldest is to use the power series expansion of atan(x)
     atan(x)=x-x^3/3+x^5/5-... together with formulas like
     pi=16*atan(1/5)-4*atan(1/239). This gives about 1.4 decimals per term.

     2) A second is to use formulas coming from Arithmetic-Geometric mean
     computations. A beautiful compendium of such formulas is given in the
     book of Borwein and Borwein: Pi and the AGM, Canadian Math. Soc. Series,
     John Wiley and Sons, New York, 1987. They have the advantage of converging
     quadratically, i.e. you double the number of decimals per iteration.
     For instance, to obtain 1 000 000 decimals, around 20 iterations are
     sufficient. The disadvantage is that you need FFT type multiplication
     to get a reasonable speed, and this is not so easy to program.

     3) A third one comes from the theory of complex multiplication of elliptic
     curves, and was discovered by S. Ramanujan. This gives a number of
     beautiful formulas, but the most useful was missed by Ramanujan and
     discovered by the Chudnovsky's. It is the following (slightly modified
     for ease of programming):

     Set k1=545140134;k2=13591409;k3=640320;k4=100100025;k5=327843840;k6=53360;

     Then in AmsTeX notation

     $\pi=\frac{k6\sqrt(k3)}{S}$, where

     $$S=\sum_{n=0}^\infty (-1)^n\frac{(6n)!(k2+nk1)}{n!^3(3n)!(8k4k5)^n}$$

     The great advantages of this formula are that

     1) It converges linearly, but very fast (more than 14 decimal digits per
     term).

     2) The way it is written, all operations to compute S can be programmed
     very simply since it only involves multiplication/division by single
     precision numbers. This is why the constant 8k4k5 appearing in the
     denominator has been written this way instead of 262537412640768000.

     This is how the Chudnovsky's have computed several billion decimals.

     Question: how can I get a C program which computes pi?

     Answer: if you are too lazy to use the hints above, you can use the
     following 160 character C program (who is the author of this?) which
     computes pi to 800 decimal digits. If you want more, it is easy to modify,
     but you have to understand how it works first.

     int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;
     for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,
     f[b]=d%--g,d/=g--,--b;d*=b);}




    References :

    (This is a short version for a more comprehensive list contact
    Juhana Kouhia at [e-mail])

    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

    David Chudnovsky and Gregory Chudnovsky
    The computation of classical constants, Columbia University,
    Proc. Natl. Acad. Sci. USA, Vol. 86, 1989.

    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



5Q:  Does there exist a number that is perfect and odd?

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



Copyright Notice

Copyright (c) 1993   A. Lopez-Ortiz


--------------------------------------------------------------------------
Questions and Answers Edited and Compiled by:

Alex Lopez-Ortiz                              [e-mail]
Department of Computer Science                      University of Waterloo
Waterloo, Ontario                                                   Canada
--
Alex Lopez-Ortiz                             [e-mail]
Department of Computer Science                      University of Waterloo
Waterloo, Ontario                                                   Canada