Different Approaches to the Distribution of Primes

In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zeta-function.     

On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g., for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions. Received November 5, 1998, Accepted December 7, 2000.     

Natural exact covering systems and the reversion of the Möbius series

The Ramanujan Journal - We prove that the number of natural exact covering systems of cardinality k is equal to the coefficient of $$x^k$$ in the reversion of the power series $$\sum _{k \ge 1} \mu...     

Residue races

Given a prime p and distinct non-zero integers a₁, a₂,..., a_k (mod p), we investigate the number N = N(a₁, a₂,..., a_k; p) of residues n (mod p) for which (na₁)_p < (na₂)_p < ... < (na_k)_p, where (b)_p is the least non-negative residue of b (mod p). We give complete solutions to the problem when k = 2,3,4, and establish some general results corresponding to k≥5. The first author is a Presidential Faculty Fellow. He is also supported, in part, by the National Science Foundation. 2000 Mathematics Subject Classification Primary—11A07; Secondary—11F20     

On complementary decompositions of the complete graph

We consider decompositionsK _n H, whereH is eitherP ₃ (the path with 3 edges) or the complete bipartite graphK _{1, 3}, with the property that upon taking the complement of each graph in the decomposition one obtains a new decompositionK _n H ^c .Research supported in part by an NSERC postgraduate Scholarship.     

Aurifeuillian factorization

The Cunningham project seeks to factor numbers of the form with small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially factored by replacing by a polynomial in such a way that polynomial factorization is possible. For example, by substituting into the polynomial factorization we can partially factor . In 1962 Schinzel gave a list of such identities that have proved useful in the Cunningham project; we believe that Schinzel identified all numbers that can be factored by such identities and we prove this if one accepts our definition of what ``such an identity' is. We then develop our theme to similarly factor for any given polynomial , using deep results of Faltings from algebraic geometry and Fried from the classification of finite simple groups.