Lawrence Somer

On a Connection of Number Theory with Graph Theory

We assign to each positive integer n a digraph whose set of vertices is H = {0, 1, ..., n − 1} and for which there is a directed edge from a ∈ H to b ∈ H if a2 ≡ b (mod n). We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.

On symmetric digraphs of the congruence x k ≡y(modn)

We assign to each pair of positive integers n and k>=2 a digraph G(n,k) whose set of vertices is H={0,1,...,n-1} and for which there is a directed edge from a@?H to b@?H if a^k=b(modn). The digraph G(n,k) is symmetric of order M if its set of components can be partitioned into subsets of size M with each subset containing M isomorphic components. We generalize earlier theorems by Szalay, Carlip, and Mincheva on symmetric digraphs G(n,2) of order 2 to symmetric digraphs G(n,k) of order M when k>=2 is arbitrary.

Manifestations of dark energy in the solar system

We give more than 10 examples based on astronomical observations showing that dark energy acts not only on large scales but also on small scales. In particular, we present several independent arguments that the average Earth-Sun distance increases by about 5 m/yr. Such a large recession speed cannot be explained by the solar wind, tidal forces, plasma outbursts from the Sun, or by the decrease of the Solar mass due to nuclear reactions. We also discuss possible reasons for disagreement with other authors, who propose much smaller values.

STABILITY OF SECOND-ORDER RECURRENCES MODULO p r

The concept of sequence stability generalizes the idea of uniform distribution. A sequence is p-stable if the set of residue frequencies of the sequence reduced modulo p r is eventually constant as a function of r . The authors identify and characterize the

A Criterion For Stability of Two-Term Recurrence Sequences Modulo Odd Primes

Consider the two-term recurrence sequence {u n} defined by u o = 0, u 1 = 1 and for all n ≥ 2, where a and b are fixed (rational) integers. Let p be a fixed odd prime such that

Generalizations of Fermat Numbers

p{\text{X}}ab(a2 + 4b)

Fundamentals of Number Theory

(1.1) . Let ξ be a root of f(x) = x 2 - ax - b in its splitting field K over Q. Let be the ring of algebraic integers in K and P a prime ideal of R lying over (p) in Z. By our assumption (1.1) on p, p is unramified.

Factors of Fermat Numbers

The current standard cosmological model is based on the normalized Friedmann equation 1 = ΩM +ΩΛ +ΩK, where ΩM is the mass density of dark and baryonic matter, ΩΛ the vacuum energy density, and ΩK is the curvature parameter. We show that the Friedmann equation was derived under excessive extrapolations from Einstein’s equations, which are not scale invariant and are “verified” on much smaller scales. We explain why these extrapolations are incorrect, why the unrestricted use of the term “verified” is questionable, and why dark matter may exist only by definition.