User jeff breeding - MathOverflow

Cubic polynomial mapping primes to primes

2011-05-05T02:15:11Z

Let $f(n)=a_3n^3+a_2n^2+a_1n+a_0$, with $a_i\in\mathbb{Z}$, $a_3>0, a_0\neq 0$ such that $f(n)>0$ for all positive integers $n$.

Given a prime $p$, when is $f(p)$ again prime?

For example, let $f(n)=7n^3-50n+30$. Then, $$f(7)=2081\quad {\rm (prime)},$$ $$f(11)=19\cdot463,$$ $$f(13)=14759\quad {\rm (prime)}.$$

Are there conditions on the $a_i$'s that guarantee that $f(p)$ is prime for all primes $p$?

Answer by Jeff Breeding for What matrix groups can be embedded in $Sp_4$?

2011-04-07T05:20:05Z

You can embed the direct product of two copies of $SL_2(\mathbb{R})$. One embedding sends the entries to the center $2\times2$ block, the other sends the entries to the corners with the $2\times2$ identity matrix in the center block. Here, the $J$ matrix is the standard anti-diagonal matrix

For embeddings of other groups, you could look at the Bruhat decomposition of Sp(4) and write a decomposition of each cell. Some explicit information is given on the decomposition for GSp(4) in a book by Ralf Schmidt and Brooks Roberts, which is available on Ralf's website.

Steinberg Representations of Finite Groups of Lie Type

2010-03-11T04:26:28Z

Let G be a finite group of Lie type. Assume G is also of universal type. Is the Steinberg representation of G generic, i.e., does the Steinberg representation admit a Whittaker model?

A Whittaker model for a representation of G is defined in a similar fashion as in the case of GL(2, F) in Bump's "Automorphic Forms and Representations." I am interested in the genericity of the Steinberg representation of a group of matrices over a finite field.

Answer by Jeff Breeding for Which mathematicians have influenced you the most?

2010-03-26T01:12:07Z

Benedict Gross. I saw him lecture a few times on BSD. His enthusiasm and mastery were very inspirational. It reminded me why I want to be a professional mathematician. I had just finished my general exams the previous semester and felt tired from taking so many classes and preparing for exams. It had put a haze over the beauty of mathematics. Professor Gross made it clear again.