Progress on Two Unsolved Problems

Some news on two of the unsolved problems previously mentioned in this blog. See Seven Mathematical Tidbits by Eric W. Weisstein and Ed Pegg Jr. for more details.

Conjecture: There are no odd perfect numbers.

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 10^300. 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 a certain composite 301-digit number can be factored.

References:
Odd Perfect Numbers by K. G. Hare
Some Factorizations That I Want by K. G. Hare

Riemann Hypothesis: All of the non-trivial zeros of the Zeta Function are located on the line Re(s) = ½.

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 not sufficient.

References:
Computation of Zeros of the Zeta Function by Xavier Gourdon and Patrick Demichel
The 10^13 First Zeros of the Zeta Function, and Zeros Computation at Very Large Height by Xavier Gourdon

Second black hole found at the centre of our Galaxy

Just three years ago, astronomers confirmed that the Milky Way revolves around a supermassive black hole, called Sagittarius A*, which is about 2.6 million times more massive than the Sun. But now a much smaller black hole, just 1,300 times our Sun's mass, has been found orbiting about three light years away from its supermassive cousin.

"This is the first intermediate-mass black hole found in our Galaxy." - Jean-Pierre Maillard, an astronomer from the Institute of Astrophysics in Paris, France.

[Thanks to DVD for providing this link.]

An interesting discussion of black holes with special emphasis on the famous Cygnus X-1 black hole candidate can be found here.

Differential Equations & Linear Algebra

With all the questionable innovations which are accepted in mathematics education whole-heartedly, there is one innovation in undergraduate mathematics which seems natural and extremely useful but has never caught on ... the integration of differential equations and linear algebra for college freshmen and sophomores after the completion of the introductory calculus sequence.

While there are many excellent textbooks for both of these subjects, there are obvious synergies between the two topics which can only be exploited by combining their teaching into one course. In fact, many professors believe (and I agree) that these two subjects must be integrated if they are to be properly understood since their intricate interaction is where all of the action is.

Some of the excellent texts which take this integrated approach are:
Introduction to Linear Algebra and Differential Equations by John Dettman
Differential Equations and Linear Algebra by Edwards & Penney (includes a lot of Matlab stuff)
Differential Equations and Linear Algebra by Jerry Farlow et al
Linear Algebra and Differential Equations by Peterson & Sochacki
Linear algebra & differential equations: An integrated approach by Charles Cullen