## Peter Cho-Ho Lam

Department of Mathematics
Simon Fraser University
Burnaby, BC V5A 1S6

Email: chohol (at) sfu (dot) ca

Last Updated: Sep 6, 2018

I am currently a fourth-year Ph.D. student in the Department of Mathematics at Simon Fraser University. My supervisors are Stephen Choi and Peter Borwein.

I completed my M.Sc. in the Department of Mathematics in August 2014 at University of Hong Kong, under the supervision of Kai Man Tsang and Yuk Kam Lau. The title of my master thesis is "Primes of the form $$x^2+Dy^2$$".

I completed my B.Sc. in the Department of Mathematics in August 2012 at Chinese University of Hong Kong.

Here is my curriculum vitae.

### Research Interests

My research focuses on analytic number theory, especially on problems related to primes. Below are some of the problems I am interested in:

1) Given two irreducible binary forms $$F, G\in\mathbb{Z}[X, Y]$$, are there infinitely many $$x, y\in\mathbb{Z}$$ such that both $$F(x, y), G(x, y)$$ are primes?

2) Let $$n\in\mathbb{N}$$ with $$n\equiv4\pmod {24}$$. Show that it can be represented as a sum of four squares with very few prime factors.

3) Distribution of Gaussian primes.

4) Variants of Titchmarsh divisor problems, e.g. $$\sum_{p\le x}\tau(p^2+1)$$

### Publications

• (With Damaris Schindler, Stanley Xiao) On prime values of binary quadratic forms with a thin variable, in preparation.
• Primes of the form $$\alpha x^2+\beta xy+\gamma y^2$$, submitted for publication.

Abstract. Let $$F(x, y)=\alpha x^2+\beta xy+\gamma y^2$$ be a positive definite and irreducible binary quadratic form and $$G(x, y)=ax+by$$ be a linear form with $$(a, b)=1$$. We prove that there are infinitely many $$x, y\in\mathbb{Z}$$ so that $$G(x, y)$$ is a prime and $$F(x, y)$$ is a prime or product of two primes. For the special case $$F(x, y)=x^2+Dy^2$$ and $$G(x, y)$$ where $$D$$ is a positive integer, we prove that they can be simultaneously prime infinitely many often.

• (With Tsz Ho Chan, Stephen Choi) Divisibility on the Sequence of Perfect Squares Minus One: the Gap Principle, published in Journal of Number Theory, Volume 184, March 2018, p. 473-484

Abstract. In this paper, we consider a gap principle when $$a^2-1|b^2-1|c^2-1$$ with $$1 < a < b < c$$. As a byproduct, we are led to determine the complete set of pairs of positive integers $$1\le u\le v\le x$$ such that $$u|v^2-1$$ and $$v|u^2-1$$ and the diophantine equation $$u^2+v^2-1=muv$$. We also generalize our main theorems to the polynomial $$f(n)=A(n+B)^2+C$$.

• Representation of Integers Using $$a^2+b^2-dc^2$$, published in J. Integer Seq., 18 (2015), Article 15.8.6.

Abstract. A positive integer $$d$$ is called special if every integer $$m$$ can be expressed as $$a^2+b^2-dc^2$$ for some nonzero integers $$a, b, c$$. A necessary condition for special numbers was recently given by Nowicki, and in this paper we prove its sufficiency. Thus, we give a complete characterization for special numbers.