TEST YOUR C.Q. [Conjecture-Quotient] Doron ZEILBERGER^{1}

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 exconjectures, 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 15), some are fairly famous (like 13 and 14), but some are really obscure (like 6 and 22). Good luck! 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 zeilberg@math.temple.edu. The top three scorers [if any] will receive each a package of M&Ms. I. The CONJECTURES

1. You can square a circle.

2. You can duplicate a cube and trisect an angle.

3. Every even integer is the sum of two primes.

4. Every integer can be written as a sum of k^{th} powers (for any k).

5. 2^{2^{n}} + 1 is always prime.

6. x^{n} + y^{n} = z^{n} and n > 2 imply that xyz = 0.

7. There do not exist non-zero integers A, B, C, D such that A^{4} + B^{4} + C^{4} = D^{4}.

8. There do not exist Greek-Latin squares of order 2 + 4k for all k ≥ 2.

1 Department of Mathematics, Temple University, Philadelphia, PA 19122. zeilberg@math.temple.edu ;

http://www.math.temple.edu/~zeilberg . Jan. 2, 1998. Exclusive for TUMM (Temple University Mathematical Monthly) and the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger.

9. Every positive-definite real polynomial, in any number of variables, can be written as a sum of squares of rational functions.

10. A simply connected n−dimensional closed manifold is homeomorphic to an n−sphere.

11. Analytic Index=Topological Index.

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.

13. The number of n × n Alternating Sign Matrices is [1!4!7! … (3n − 2)!]/[n!(n + 1)! … (2n − 1)!].

14. The coefficient of x_{1}^{0} … x_{n}^{0} in the Laurent polynomial

∏_{1≤i<j≤n}{∏_{r=0}^{a_{i}}(1 − q^{r}x_{i}/x_{j}) ∏_{r=1}^{a_{j}+1}(1 − q^{r}x_{j}/x_{i})}

equals [a_{1} + … a_{n}]!/([a_{1}]! … [a_{n}]!), where [m]! = (1)(1 + q)(1 + q + q^{2}) … (1 + q + … q^{m−1}).

15. For every integer k ≥ 0, the Taylor coefficients of (1 − q)^{k}/[(1 − q)(1 − q^{2}) … (1 − q^{k})] at q = 0 alternate in sign (weakly, i.e. 0 can be taken to be positive or negative).

16. ∑_{n=1}^{∞} n^{−3} is irrational.

17. Every set in R^{d} is a union of d + 1 sets of smaller diameter.

18. You can hear the shape of a drum (i.e. the spectrum of the Dirichlet problem on a plane region determines its shape).

19. Any univalent function f on the unit disc such that f(0) = 0 and f′(0) = 1 has |f^{(n)}(0)/n!| ≤ n.

20. Every planar map is 4-colorable.

21. There is nothing between ℵ_{0} and the cardinality of R.

22. All non-trivial zeros of the meromorphic continuation of ∑_{n=1}^{∞} n^{−z} have Rez = 1/2.

23. Every finitely generated projective module over k[t_{1}, …, t_{n}] is free.

24. There is a decision-procedure to decide whether any given diophantine equation has solutions.

25. Let τ(n) be the coefficient of q^{n} in q(1 − q)^{24}(1 − q^{2})^{24}(1 − q^{3})^{24} … then |τ(n)| ≤ d(n)n^{11/2}, where d(n) is the number of divisors of n.

26. The analog of 22 over finite fields.

27. For any root system R the constant term of ∏_{α∈R}(1 − x^{α})^{a} is ∏_{i=1}^{l} \binom{d_{i}a}{a}, where d_{1}, …, d_{l} are the fundamental invariants of R.

28. If the section function of a centered convex body in E^{n}, n ≥ 4 is smaller than that of another such body, then its volume is also smaller.

29. The class of NP functions strictly contains the class of P functions.

30. A finite group of odd order is solvable.

31. Every differential operator with constant coefficients has a fundamental solution.

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).

33. The complement of a perfect graph is perfect.

34. The permanent of an n × n doubly-stochastic matrix is ≥ n!/n^{n}.

35. 2^{\sqrt{2}} is transcendental.

36. The number of totally symmetric plane partitions with largest part ≤ n is equal to

∏_{1≤i≤j≤k≤n} \frac{i + j + k − 1}{i + j + k − 2} .

37. The number of 2-stack-sortable permutations on n objects is 2(3n)!/((2n + 1)!(n + 1)!).

38. The number of rational points of an algebraic curve C defined over Q with genus larger than 1 is finite.

39. The eignevalues of the Hecke operators for Mass forms of Galois type are algebraic.

40. Euler’s constant γ := lim_{n→∞} [(∑_{i=1}^{n} 1/i) − log n] is irrational.

41. No packing of spheres in Euclidean 3−space has density exceeding that of the face-centered cubic lattice packing, which is π/\sqrt{18} = 0.74048 · · ·.

42. β := lim_{n→∞} 2nE_{2n}(|x|), (where E_{2n}(f) is the error in the best L^{∞}[−1, 1] approximation of f by polynomials of degree ≤ 2n) is equal to \frac{1}{2\sqrt{π}}. 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. Poincaré; Srinivasa Ramanujan; Bernard Riemann; Jean-Pierre Serre; Richard Stanley; Bartel L. van der Waerden; Edward Waring; André Weil; Julian West.

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 Lamé[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].

IV. The PROVERS and DISPROVERS

Roger Apéry; 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; Lászo Lovász; 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.

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.