MathWorld Headline News

Seven Mathematical Tidbits

By Eric W. Weisstein and Ed Pegg Jr.

November 8, 2004--While the last several months have not been filled with any particularly earth-shattering new mathematical results, a number of interesting events, findings, and mathematical books have recently appeared. Here is a recap of some of them.

1. Martin Gardner Celebrates His 90th

Martin Gardner, well-known mathematical essayist, popularizer, and author of Scientific American's long-running Mathematical Games column, celebrated his 90th birthday on October 21, 2004. The Mathematical Association of America has a birthday greeting. Gardner's collected essays fill more than a dozen books, many of which have been extensively mined on MathWorld.

2. Odd Perfect Numbers Must Be Odd Indeed

A perfect number is a positive integer n such that n = s(n), where s(n) is the sum of proper divisors of n. For example, the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. Perfect numbers were deemed to have important numerological properties by the ancients, and were extensively studied by the Greeks, including Euclid. While many of Euclid's successors implicitly assumed that all perfect numbers were even, the precise statement that all even perfect numbers are of this form was first considered in a 1638 letter from Descartes to Mersenne. Today, it is still not known if any odd perfect numbers exist, although it is known that any such number would have to be greater than 10³⁰⁰.

Mathematician Kevin Hare has been investigating the existence of odd perfect numbers, and has proven that any odd perfect number must have at least 47 prime factors (including repetitions), and that improving this result depends upon finding factors of three large numbers. In particular, a higher bound can be computed if the composite 301-digit number can be factored.

3. 10 Trillion Zeta Zeros

The Riemann zeta function is a number theoretical function defined on the complex plane that also finds extensive use in mathematical physics. Its properties turn out to be intimately connected with the distribution of prime number, and as a result it has received extensive study.

Numerical experiments indicate that the "nontrivial" zeros of the Riemann zeta function lie along the so-called critical line, i.e., the line corresponding to real part x = 1/2. One of the most difficult problems in mathematics seeks to prove that all nontrivial zeros lie along this line, a proposition known as the Riemann hypothesis. There is even a $1 million prize for a proof of this hypothesis (although curiously, the prize is not offered for a disproof of the hypothesis).

On October 13, Xavier Gourdon and Patrick Demichel announced that they had used an efficient technique due to Odlyzko and Schönhage to find the first ten trillion nontrivial zeroes of the Riemann zeta function, more than ever before computed. Every single one of these zeros lies along the critical line, which is a necessary requirement for the Riemann hypothesis to hold, but regrettably not sufficient.

4. Mathematica GuideBooks for Programming and Graphics

  

The Mathematica GuideBooks are an exploration of Mathematica's application to classic and current problems in science, computer science, and visualization by physicist and Mathematica expert Michael Trott of Wolfram Research. The first two of a planned four volumes have just been released by Springer-Verlag.

The Mathematica GuideBook for Programming provides a comprehensive, step-by-step development of Mathematica's programming capabilities and contains an enormous collection of examples and worked exercises. The The Mathematica GuideBook for Graphics provides the same on how to use Mathematica to visualize functions and data, manipulate graphics, and optimize their appearance. Two-dimensional graphics, contour plots, plots of surfaces, free-form three-dimensional surfaces, and animations are the core topics. Both volumes contain all the executable code and outputs and are provided on an accompanying DVD.

The Numerics and Symbolics volumes are scheduled to be published Spring 2005. (The DVD-ROMs which come with the Programming and Graphics volumes contain the text and inputs from all four volumes, but not outputs, graphics, or animations.) Michael's site mathematicaguidebooks.org, has extensive supporting material for these new books.

The Mathematica GuideBooks are true mathematical gems. Overflowing with beautiful results, extensive literature references, and stunning graphics, these books provide a fascinating glimpse into the power of computational mathematics. Michael Trott's expert knowledge of the Mathematica programming language make these books an indispensable reference to both novice and experienced Mathematica programmers, and his encyclopedic knowledge of math, physics, and the literature make these books a mathematical tour de force. The authors have no doubt that the GuideBooks will rapidly become among the most treasured books in the libraries of students, researchers, and math enthusiasts alike.

5. 3rd Edition of UPINT

The third edition of Richard Guy's classic text Unsolved Problems in Number Theory has been released by Springer-Verlag. Fondly known as "UPINT", Guy's book has remained an up-to-date and valuable resource for the state of the art in number theory through its previous three editions and know through the current one. In fact, the second edition has been extensive mined in MathWorld, which contains more than 300 detailed references to the work.

The third edition contains many updates. In fact, if MathWorld readers spot any information that needs updating, please report it to mathworld@wolfram.com so we can incorporate the latest results. In addition, in the third edition Guy has added extensive links to N.J.A. Sloane's Online Encyclopedia of Integer Sequences (OEIS).

6. Online Encyclopedia of Integer Sequences 100K E-Party

Speaking of the Online Encyclopedia of Integer Sequences, sequence A100000 was posted on November 7. The sequence chosen for this honor is "3, 6, 4, 8, 10, 5, 5, 7", which gives the middle column of marks found on the oldest object with logical carvings, the 22000-year-old Ishango bone from the Congo.

The On-Line Encyclopedia of Integer Sequences in the brainchild and labor of love of mathematician Neil Sloane. Its main purpose is to allow mathematicians or other scientists to find out if some sequence that turns up in their research has ever been seen before. Another purpose is to have an easily accessible database of important, but difficult to compute, sequences. The on-line encyclopedia has been compiled by Neil and collaborators over the course of decades, and supersedes printed versions published in 1973 and 1995.

To celebrate attainment of the 100K mark, there is an OEIS 100K e-party. Astute readers will recognize both authors of this news story as participants in Neil Sloane's e-party. Regular MathWorld readers will also recognize that OEIS is extensively cross-linked to thousands of links is each direction.

7. Encyclopedia of Triangle Centers

A triangle center is a point whose trilinear coordinates are defined in terms of the side lengths and angles of a reference triangle and for which a so-called triangle center function can be defined. Well-known triangle centers include the incenter I, centroid G, circumcenter O, and orthocenter H, illustrated above (left figure) for the reference triangle ABC.

These are just four (in fact, the first four), of the triangle centers compiled by mathematician Clark Kimberling from the University of Evansville. 301 (plus 13 additional) centers appeared in Kimberling (1994) and 360 in Kimberling (1998), but Kimberling has continued to compile centers (called Kimberling centers in this work) online in his "Encyclopedia of Triangle Centers." This list has recently received a major update and contains a total of 2676 triangle centers (a subset of which are illustrated at right above). A subset of these centers are available in the Mathematica package KimberlingCenters.m (to be used in conjunction with PlaneGeometry.m, both written by Eric Weisstein), and trilinears for the entire ETC should be available soon as result of work by Dr. Edward Brisse in transcribing the centers into Mathematica.