University of South Carolina

Number Theory Seminar

Fall 2019

Unless otherwise indicated, the seminar will be in LeConte 317R from 1:30 p.m. to 2:20 p.m.

If you are interested in giving a talk contact Joe Foster at josephcf@math.sc.edu.

On the irreducibility of the non-reciprocal part of polynomials of the form $$f(x)x^n+g(x)$$

A recent paper of William Sawin, Mark Shusterman and Michael Stoll introduces the notion of robust pairs of polynomials in $\mathbb{Z}[x]$ and shows that under a condition of robustness the polynomial f (x)xⁿ + g (x) has an irreducible non-reciprocal part provided n is larger than an explicit bound depending only on f (x) and g (x). In this talk, I discuss recent work with Huixi Li, Frank Patane and Dane Skabelund on an improved bound.

What do Michael's students do?

The number $p=1289$ is prime. The polynomial $f(x)=x^3+2x^2+8x+9$ (which satisfies $f(10)=p$) is irreducible. A theorem attributed to A. Cohn asserts that this is no coincidence and that all polynomials arising from primes in this way will be irreducible. Work of M. Filaseta and others has extended this result to include relaxations on the coefficients being digits in the range $[0,9]$ but to $[0,167]$ and even numbers for which this abstract is too narrow to contain. Filaseta's current trio of buffoons are working on extending this result to show that there exist bounds on the coefficients of non-negative, integer polynomials that assure irreducibility if the polynomial takes on a prime value. In this talk I will attempt to not embarrass myself whilst updating you on the progress of this problem (Spoiler: we think we've done it). The other two buffoons are Jacob Juillerat and Jeremiah Southwick.

The infinitude of Weak Primes

Define a weak prime to be a prime with the property that if you change any one of its digits to some other digit, the resulting number is composite. Our discussion will be confined to primes that are weak base 10, but the notation makes sense in any base. We will give Erdos' 1979 proof that the set of such primes is infinite. If time allows, we will delve into other results in the literature related to weak primes.

No talk this week.

No talk this week.

Estimating the number of points near a closed convex curve by curvature.

Let $C$ be a close curve in the plane and assume that the radius of curvature, $\rho$, of this curve has bounds

Then we show that there is a explicitly computed number $\delta$, only depending on $R_1$ and $R_2$ so that the number of lattice points at a distance less than $\delta$ from $C$ is at most $6L/R_1^{1/3}$ where $L$ is the length of $C$. The proof only uses elementary methods from differential geometry and number theory.

This is joint work with Ognian Trifonov.

Klingen vectors of level 2 for $GSp(4)$.

In 2007, Roberts and Schmidt had a satisfactory local new- and oldform theory for $GSp(4)$ with trivial central character, in which they considered the vectors fixed by the paramodular groups $K(p^n)$. In this talk, we consider the space of vectors fixed by the Klingen subgroup of level $p^2$. We determine the dimensions of the spaces of these invariant vectors for all irreducible, admissible representations of $GSp(4)$ over a $p$-adic field. The results can be applied to determine the dimension formula for Siegel modular forms of degree 2 with Klingen level 4, which is an ongoing project with Ralf Schmidt.

A first lecture in algebraic Number Theory

At an earlier time, when we didn't have the faculty with the interests of our current faculty, I conceived of the idea of teaching a course on "algebraic Number Theory" different from courses I had taken on "Algebraic number theory" where the emphasis in "algebraic Number Theory" would be on using algebraic concepts to resolve Number Theoretic problems rather than creating Algebraic concepts reflecting number theoretic notions. Glancing at some notes written by David Richman, I decided to steal ideas of his for a first lecture. Now, years later, after my PhD students have completed a course in Algebraic number theory and finding that they don't know algebraic Number Theory and that they need this first lecture to understand an aspect of their dissertation work, I feel compelled to give David's first lecture again, introducing Number Theory in action via a simple algebraic concept.

Super Digitally Delicate Primes

Filaseta, Kozek, Nicol, and Selfridge showed in 2011 how covering systems can be leveraged to produce an infinite number of composite numbers which remain composite if a single digit in their decimal representation is replaced by some other digit. We present this result and show how it can be modified, along with ideas of Erdos and Tao, to show a positive proportion of prime numbers become composite if any digit in their decimal representation - including any of the infinitely many leading 0’s - is changed to some other digit. This work is joint with Michael Filaseta.

We did (part of) another thing. Here is that part

The speaker has found that the best way to get his adviser thinking about the speaker's research is to present a talk on said research. With this goal in mind, the speaker will talk about a thing we did part of - something involving a sequence of polynomial sequences and irreducibility. The we is Joe Foster and the speaker, but not Jacob Juillerat.

No talk this week.

On sums of fractional parts

This a survey talk on recent progress on estimating exponential sums, and on some of the ideas that went into it.

Previous Seminars