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