On the prime power factorization of n!

In this paper we prove two results. The first theorem uses a paper of Kim \cite{K} to show that for fixed primes $p_1,...,p_k$, and for fixed integers $m_1,...,m_k$, with $p_i\not|m_i$, the numbers $(e_{p_1}(n),...,e_{p_k}(n))$ are uniformly distributed modulo $(m_1,...,m_k)$, where $e_p(n)$ is the order of the prime $p$ in the factorization of $n!$. That implies one of Sander's conjecture from \cite{S}, for any set of odd primes. Berend \cite{B} asks to find the fastest growing function $f(x)$ so that for large $x$ and any given finite sequence $\epsilon_i\in \{0,1\}, i\le f(x)$, there exists $n<x$ such that the congruences $e_{p_i}(n)\equiv \epsilon_i\pmod 2$ hold for all $i\le f(x)$. Here, $p_i$ is the $i$th prime number. In our second result, we are able to show that $f(x)$ can be taken to be at least $c_1 (\log x/(\log\log x)^6)^{1/9}$, with some absolute constant $c_1$, provided that only the first odd prime numbers are involved.Comment: 7 pages; accepted Journal of Number Theor

Synthesizing Imperative Programs from Examples Guided by Static Analysis

We present a novel algorithm that synthesizes imperative programs for introductory programming courses. Given a set of input-output examples and a partial program, our algorithm generates a complete program that is consistent with every example. Our key idea is to combine enumerative program synthesis and static analysis, which aggressively prunes out a large search space while guaranteeing to find, if any, a correct solution. We have implemented our algorithm in a tool, called SIMPL, and evaluated it on 30 problems used in introductory programming courses. The results show that SIMPL is able to solve the benchmark problems in 6.6 seconds on average.Comment: The paper is accepted in Static Analysis Symposium (SAS) '17. The submission version is somewhat different from the version in arxiv. The final version will be uploaded after the camera-ready version is read

Density of rational points on isotrivial rational elliptic surfaces

For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We also prove that these surfaces satisfy a variant of weak-weak approximation. Our results are conditional on the finiteness of Tate-Shafarevich groups for elliptic curves over the field of rational numbers.Comment: Latex; 26 pages. To appear in Algebra and Number Theor

Automatic sets of rational numbers

The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.Comment: Previous version appeared in Proc. LATA 2012 conferenc