##### Department of Mathematics,

University of California San Diego

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### Math 196/296 - Student Colloquium

## Jamie Pommersheim

#### Reed College

## Is infinity plus one prime? (Or: Factorization in the omnific integers)

##### Abstract:

In the 1970s, John Horton Conway introduced the {\sl surreal numbers}, a number system that contains not only the real numbers, but contains many infinite numbers as well. Among the elements of this field, one finds $\omega$ (the first infinite ordinal), $\omega + 1$ (``infinity plus one''), $\omega - 1$ (``infinity minus one''), $\frac{1}{\omega}$ (an infinitesimal) and as well as numbers like $\frac{\omega^{\omega}}{\sqrt{\omega^2 - \pi}} + \omega^{\frac{1}{\omega}}$. Within the surreal numbers, there is also a natural generalization of the integers, called the {\sl omnific integers}, denoted $\mathbb{O}\mathbb{Z}$. The first three numbers listed above are omnific integers, while the last two are not. With this new notion of integer, one can revisit many of the classical questions of number theory. Most immediately, what are the prime numbers? Can every omnific integer be factored uniquely into omnnific prime numbers? In this talk, after introducing the surreal numbers, we'll explore some questions about primes in the land of $\mathbb{O}\mathbb{Z}$.

Host: David Meyer

### October 26, 2006

### 12:00 PM

### AP&M B402A

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