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\bf
\centerline{ TEST YOUR C.Q.
[Conjecture-Quotient]}
\medskip
\it
\centerline{Doron ZEILBERGER\footnote{$^1$}
{\baselineskip=9pt
\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122.
{\baselineskip=9pt
\eighttt zeilberg@math.temple.edu ; 
\break http://www.math.temple.edu/\Tilde 
zeilberg }.
Jan. 2, 1998.
Exclusive for TUMM (Temple University Mathematical Monthly)
and the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger.
}}
\medskip
\rm
 
{\bf PREFACE.} According to Gian-Carlo Rota, Richard Feynman's secret to
being a genius is to be aware of many open problems, and
everytime you learn a new trick, try it out on all them.
Most of the time, the new trick won't do you any good, but
once or twice in a lifetime, it would exactly solve one of the problems,
and then everybody would say: Gee, what a genius!
 
Hence according to Feynman (according to Rota), a necessary condition
for being a genius is knowing about open problems. According to
Zeilberger, it is also important to know about ex-conjectures,
and about how they were solved [this will teach you
some neat tricks]. Hence the following
trivia quiz is highly non-trivial!
 
Some of the conjectures are extremely famous (like 
{\bf 15}), some are fairly famous (like {\bf 13} and
{\bf 14}), but some are really obscure (like {\bf 6} and {\bf 22}).
Good luck!
 
{\bf INSTRUCTIONS.} To each of the statements in
section I, match the mathematician in section
II [if any] who conjectured it, the mathematician in section III,
[if any],
who believed he had a proof, but, alas, was mistaken, and
the mathematician(s) in section IV[if any] who first settled the
conjecture, [so far] for good. Also state whether
the statement is true, false, undecidable, or still open.
 
Send solutions to {\tt zeilberg@math.temple.edu}.
The top three scorers [if any] will receive 
each a package of M\&Ms.
 
{\bf I. The CONJECTURES}
 
{ \bf 1. } You can square a circle.
 
{ \bf 2.} You can duplicate a cube and trisect an angle.
 
{\bf 3. } Every even integer is the sum of two primes.
 
{\bf 4.} Every integer can be written as a sum of
$k^{th}$ powers  (for any $k$).
 
{\bf 5.} $2^{2^n}+1$ is always prime.
 
{\bf 6.} $x^n+y^n=z^n$ and $n>2$ imply that $xyz=0$.
 
{\bf 7.} There do not exist  non-zero integers $A,B,C,D$ such that
$A^4+B^4+C^4=D^4$.
 
{\bf 8.} There do not exist Greek-Latin squares of
order $2+4k$ for all $k \geq 2$.
 
{\bf 9.} Every positive-definite real polynomial, in
any number of variables,  can be
written as a sum of squares of rational functions.
 
{\bf 10.} A simply connected $n-$dimensional closed manifold
is homeomorphic to an $n-$sphere.
 
{\bf 11.} Analytic Index=Topological Index.
 
{\bf 12.} If the Jacobian determinant of a polynomial
transformation from $n$-space to $n-$space is identically
[a non-zero] constant, then the inverse mapping is also polynomial.
 
{\bf 13.} The number of $n \times n$ Alternating Sign Matrices
is $[1!4!7! \dots (3n-2)!]/[n!(n+1)! \dots (2n-1)!]$.
 
{\bf 14.} The coefficient of
$x_1^0 \dots x_n^0$ in the Laurent polynomial
$$
\prod_{1 \leq i < j \leq n}
\{\prod_{r=0}^{a_i} (1-q^r x_i/x_j)
\prod_{r=1}^{a_j+1} (1-q^r x_j/x_i) \}
$$
equals $[a_1 + \dots a_n]!/([a_1]! \dots [a_n]!)$,
where $[m]!=(1)(1+q)(1+q+q^2) \dots (1+q+ \dots q^{m-1})$.
 
{\bf 15.} For every integer $k \geq 0$, the Taylor coefficients of
$(1-q)^k/[(1-q)(1-q^2) \dots (1-q^k)]$ at $q=0$ alternate
in sign  (weakly,
i.e. 0 can be taken to be positive or negative).
 
{\bf 16.} $\sum_{n=1}^{\infty} n^{-3}$ is irrational.
 
{\bf 17.} Every set in $R^d$ is a union of
$d+1$ sets of smaller diameter.
 
{\bf 18.} You can hear the shape of a drum
(i.e. the spectrum of the Dirichlet problem on
a plane region determines its shape).
 
{\bf 19.} Any  univalent function $f$ on the unit disc
such that $f(0)=0$ and $f'(0)=1$ has
$\vert f^{(n)}(0)/n! \vert \leq n$.
 
{\bf 20.} Every planar map is 4-colorable.
 
{\bf 21.} There is nothing between $\aleph_0$ and
the cardinality of $R$.
 
{\bf 22.} All non-trivial zeros of the meromorphic
continuation of
$\sum_{n=1}^{\infty} n^{-z}$ have $Re z=1/2$.
 
 
{\bf 23.} Every finitely generated projective module
over $k[t_1, \dots , t_n]$ is free.
 
{\bf 24.} There is a decision-procedure to decide
whether any given diophantine equation has solutions.
 
{\bf 25.} Let $\tau(n)$ be the coefficient of
$q^n$ in $q(1-q)^{24}(1-q^2)^{24} (1-q^3)^{24} \dots $ then
$\vert \tau(n) \vert \leq d(n) n^{11/2}$, where
$d(n)$ is the number of divisors of $n$.
 
{\bf 26.} The analog of {\bf 22} over finite fields.
 
{\bf 27.} For any root system $R$ the constant term
of  $\prod_{\alpha \in R} (1-x^{\alpha})^a$ is
$\prod_{i=1}^{l} {{d_i a} \choose {a} }$, where
$d_1, \dots, d_l$ are the {\it fundamental invariants}
of $R$.
 
{\bf 28.} If the section function of a 
centered convex body in $E^n$, $n \geq 4$ is smaller than
that of another such body, then its volume is also smaller.
 
{\bf 29.} The class of NP functions {\it strictly} contains the
class of P functions.
 
{\bf 30.} A finite group of odd order is solvable.
 
{\bf 31.} Every differential operator with constant coefficients
has a fundamental solution.
 
{\bf 32.} The general solution of an arbitrary system
of linear partial differential equations with constant coefficients
can be expressed as an ` infinite linear combination' of
exponential-polynomial solution (expressed as integrals over
multiplicity varieties).
 
{\bf 33.} The complement of a perfect graph is perfect.
 
{\bf 34.} The permanent of an $n \times n$ doubly-stochastic
matrix is $\geq n!/n^n$.
 
{\bf 35.} $2^{\sqrt{2}}$ is transcendental.
 
{\bf 36.} The number of totally symmetric plane partitions
with largest part $\leq n$ is equal to
$$
\prod_{1 \leq i \leq j \leq k \leq n} {{i+j+k-1} \over {i+j+k-2}}
\quad .
$$
 
{\bf 37.} The number of $2$-stack-sortable permutations on
$n$ objects is $2 (3n)! /((2n+1)! (n+1)!)$.
 
{\bf 38.} The number of rational points of an algebraic curve
$C$ defined over $Q$ with genus larger than $1$ is finite.
 
{\bf 39.}  The eignevalues of the Hecke operators for Mass forms
of Galois type are algebraic.
 
{\bf 40.} Euler's constant 
$\gamma:=\lim_{n \rightarrow \infty} \left [ (\sum_{i=1}^{n} 1/i )
-\log n \right ]$
is irrational.
 
{\bf 41.} No packing of spheres in Euclidean $3-$space has density
exceeding that of the face-centered cubic lattice packing, which is
$\pi/\sqrt{18}=0.74048 \cdots$.
 
{\bf 42.} $\beta:=\lim_{n \rightarrow \infty} 2nE_{2n}(\vert x \vert)$,
(where $E_{2n}(f)$ is the error in the best $L^{\infty}[-1,1]$ 
approximation of
$f$ by polynomials of degree $\leq 2n$) is equal to
${{1} \over {2 \sqrt{\pi}}}$.
 
 
 
{\bf II. The CONJECTURERS}
 
George Andrews;   Anon.[or I don't know]
(6);  Claude Berge;  Stephen Bernstein; Ludwig Bieberbach;  
Karol Borsuk;  William S. Burnside;  Busemann-Petty;
Lehonard Euler(2);  Pierre Fermat(2) ;
Christian Goldbach;  Frederick Guthrie;
David Hilbert(5); Marc Kac; Richard Karp(?);
Keller; Johann Kepler ; 
Ian G. Macdonald; Robert Mills-David Robbins-Howard Rumsey;  
Louis J. Mordell; Andrew Odlyzko;
Oracle of Delphi(2);
Henry J. Poincar\'e; Srinivasa Ramanujan; Bernard Riemann;
Jean-Pierre Serre; Richard Stanley; Bartel L. van der Waerden; Edward Waring;  
Andr\'e Weil; Julian West.
 
{\bf III. The WOULD-BE PROVERS}
 
Paul Appel [published in Comptes Rendus, 1926];
D.Blasius-Laurent Clozel-D.Ramakrishnan [published in Comptes 
Rendus 1989];
 Louis de Branges[published in J. Math. Anal. Appl., 1972];
 Louis de Branges[unpublished but announced several times],
 Hans Rademacher [announced, submitted but retracted], and
 many others;
David Hilbert [published in Annalen];
 Wu-Yi Hsiang [published in Inter. J. Math, 1993];
 A.B. Kempe [published in Amer. J. Math., 1879];
 Ernst E. Kummer[submitted], Gabriel Lam\'e[published
in Comptes Rendus], C. L. Ferdinand Lindemann [published in Annalen];
McNeish[1927, published in Annals of Math, claimed (incorrectly)that conj. is 
true];
Ed Nelson [announced, but not published];
Gaoyong Zhang[published in Annals of Math., 1994].
 
 
{\bf IV. The PROVERS and DISPROVERS}
 
Roger Ap\'ery; 
 Ken Appel-Wolfgang Haken;
 Emil Artin;
 Michael Atiyah- Isadore Singer;
 Louis de Branges;
 Paul Cohen;
 Piere Deligne(2);
 G.P. Egoritsjev-D.I. Falikman;
Leon Ehrenpreis- Bernard Malgrange;
Leon Ehrenpreis- Viktor Palamadov;
 Noam Elkies;
 Lehonard Euler;
 Gerd  Faltings;
 Walter Feit and John G. Thompson; 
 Evariste Galois[essentially];
 Aleksander O. Gelfond-Theodor Schneider;
 Carolyn Gordon and David Webb;
 David Hilbert;
 Jeff Kahn and Gil Kalai;
 C.L. Ferdinand Lindemann;
 L\'aszo Lov\'asz;
 Yuri Matiyachevich;
 John Milnor ($n>$4)- Simon Donaldson, Michael Freedman ($n=4$)- 
still open($n=3$);
 Eric Opdam [after ingenious proofs of the $G_2$ case
 by Laurent Habsieger and Doron Zeilberger];
 Danile G. Quillen-Mihail J. Suslin;
 Dennis Stanton and Doron Zeilberger;
 Still Open [as far as I know](8);
 John Stembridge; 
  R.C. Bose and D.K.Ray-Chaudhuri;
 Richard S. Varga and Amos Carpenter; 
 Andrew Wiles;
 Doron Zeilberger(2);
 Doron Zeilberger and Dave Bressoud.
 
{\bf CONCLUSION.}
 
I will let you decide for yourself
whether you
are a CG (Conjecture-Genius), or CR (Conjecture-Retard),
or somewhere in between. Also note the distinguished 
set of mathematicians who made it to section III
(many of whom also show up in section IV), so
learn from Gian-Carlo Rota who said not to worry
about your mistakes.
 
 
\bye
 
 
 
 
 
 
