A seminar series.
Rick Luttmann
(Sonoma State)
Carmichael Numbers
Carmichael Numbers are composite counterexamples to the converse of Fermat's "Little Theorem". I will explain and prove the Fermat theorem, then consider its converse and why it's false. There is quite a bit known about Carmichael Numbers. It was only a quarter-century ago that it was determined there are infinitely many.
Jay Cummings, Quin Darcy and Morgan Throckmorton
(Sacramento State)
Natalie Hobson, Drew Horton, Keith Rhodewalt, and Ry Ulmer-Strack
(Sonoma State)
Counting Pseudo Progressions
Arithmetic progressions are sequences of numbers in which each consecutive term differ by the same constant. If we allow for more than one common difference between consecutive terms, then the progression is called a pseudo progression; if we allow up to m common differences, then it is called an $m$-pseudo progression. We will explore how to count the number of pseudo progressions in the set $\{1,2,\dots,n\}$, and discuss the applications of this to future work in Ramsey theory.
Morgan Throckmorton
(Sacramento State)
Coloring the Integers: Pseudo Progressions and Ramsey Theory
Van der Waerden's theorem states that if you color the integers $r$ distinct colors,
there exists a least positive integer $n$ for which every coloring of $\{1,2,...,n\}$
contains a $k$-term monochromatic arithmetic progression. For our research,
we expand on the definition of an arithmetic progression and define what we
call an $m$-pseudo progression. We then use the idea behind this theorem to study
our own van der Waerden-like numbers by looking for pseudo progressions.
In this talk, I will introduce the basic idea behind our research and then
further explain how we are using integer partitions and other methods to find
these values.