Investigations in Number Theory.

 

Number Theory, called "The Queen of Mathematics" by Gauss, is an ideal source of problems for the Polymath Jr collaboration. Many problems require no more than high school algebra to state and understand, though the methods we believe will yield full proofs often involve advanced machinery. Problems will be chosen from a variety of sub-fields, including Additive Combinatorics (especially Zeckendorf decompositions and More Sums Than Difference Sets), Benford's Law of Digit Bias (see below), elliptic curves, and low-lying zeros. Projects will frequently have many sub-problems to consider, allowing you to have a chance to work in smaller groups while also having colleagues you can talk to who are doing similar analyses. The specific problems will not be chosen until the background and interest of the participants is known. This is the model we have used for the last two Polymath Jrs.

Benford's law of digit bias states that in many data sets each digit 1 thru 9 is not equally likely to be the leading digit, but rather we observe 1 almost 30% of the time, with the probability falling down to about 4.6% for starting with a 9. In addition to being of theoretical interest, this is also used in detecting various types of fraud. We will explore both aspects. On the fraud side, one
project is to explore new tests to use Benford's law to examine whether or not there has been fraud in data sets (examples to be drawn from election and medical research). On the theory side, last summer our Polymath Jr group explored several topics, including connections with fractal sets; the
goal is to continue this program.