Model Theory Seminar

In 1964, Ulam introduced his eponymous sequence, defined as starting with two integers a and b, and such that every subsequent term is the smallest integer that can be written as a sum of two distinct prior terms in exactly one way. This sequence has mystified number theorists ever since---there are many examples of conjectures about Ulam sequences that are very well supported numerically, but seem impossible to prove due to the chaotic nature of the sequence. In this talk, we'll show that in contrast families of Ulam sequences seem to be incredibly rigid, and that by appealing to a non-standard model of arithmetic, we can actually prove some interesting results in this direction.