Number Theory Commons^™

The Probability Of Relatively Prime Polynomials, Arthur T. Benjamin, Curtis D. Bennet

All HMC Faculty Publications and Research

No abstract provided in this article.

Galois Number Fields With Small Root Discriminant, John W. Jones, David P. Roberts

Mathematics Publications

We pose the problem of identifying the set K(G,Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632. We definitively treat the cases G = A₄. A₅, A₆, and S₄, S₅, S₆, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois groups SL₃(2), A₇, S₇, PGL₂(7), SL₂(8), ΣL₂(8), PGL₂(9), PSL₂(11), and …

Two Number-Theoretic Problems That Illustrate The Power And Limitations Of Randomness, Andrew Shallue

Scholarship

This thesis contains work on two problems in algorithmic number theory. The first problem is to give an algorithm that constructs a rational point on an elliptic curve over a finite field. A fast and easy randomized algorithm has existed for some time. We prove that in the case where the finite field has characteristic 2, there is a deterministic algorithm with the same asymptotic running time as the existing randomized algorithm.

Continued Fractions With Multiple Limits, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

For integers m ≥ 2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a class of Poincar´e type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity. We also generalize a curious q-continued fraction of Ramanujan’s with three limits to a continued fraction with …

Wild Partitions And Number Theory, David P. Roberts

Mathematics Publications

We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of p-adic fields. For Q a power of p, we get a sequence of numbers λ_Q,n counting the number of certain wild partitions of n. We give an explicit formula for the corresponding generating function Λ_Q(x) = Σλ_Q,nx^_n and use it to show that λ^1/n _Q,n tends to Q^1/(p-1). We apply this asymptotic result to support a finiteness conjecture about number fields. Our finiteness conjecture …

Some Observations On Khovanskii's Matrix Methods For Extracting Roots Of Polynomials, James Mclaughlin, B. Sury

Mathematics Faculty Publications

In this article we apply a formula for the n-th power of a 3×3 matrix (found previously by the authors) to investigate a procedure of Khovanskii’s for finding the cube root of a positive integer. We show, for each positive integer α, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to α 1/3 . We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a …

Ramanujan And Extensions And Contractions Of Continued Fractions, James Mclaughlin, Nancy Wyshinski

Mathematics Faculty Publications

If a continued fraction K∞n=1an/bn is known to converge but its limit is not easy to determine, it may be easier to use an extension of K∞n=1an/bn to find the limit. By an extension of K∞n=1an/bn we mean a continued fraction K∞n=1cn/dn whose odd or even part is K∞n=1an/bn. One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii) Prove the extension converges and find the limit of the other contraction (for example, the odd part, if K∞n=1an/bn is the even part); (ii) Find the limit of the …

Symmetry And Specializability In The Continued Fraction Expansions Of Some Infinite Products, James Mclaughlin

Mathematics Faculty Publications

Let f(x) ∈ Z[x]. Set f0(x) = x and, for n ≥ 1, define fn(x) = f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product Y∞ n=0 ( 1 + 1 fn(x) ) has a specializable continued fraction expansion of the form S∞ = [1; a1(x), a2(x), a3(x), . . . ], where ai(x) ∈ Z[x] for i ≥ 1. When the infinite product and the continued fraction are specialized by letting x take integral values, we get infinite classes of real numbers whose regular continued fraction expansion is predictable. We also show that, under some …

Some More Long Continued Fractions, I, James Mclaughlin, Peter Zimmer

Mathematics Faculty Publications

In this paper we show how to construct several infinite families of polynomials D(¯x, k), such that p D(¯x, k) has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter k. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.

Some Properties Of The Distribution Of The Numbers Of Points On Elliptic Curves Over A Finite Prime Field, Saiying He, James Mclaughlin

Mathematics Faculty Publications

Let p ≥ 5 be a prime and for a, b ∈ Fp, let Ea,b denote the elliptic curve over Fp with equation y 2 = x 3 + a x + b. As usual define the trace of Frobenius ap, a, b by #Ea,b(Fp) = p + 1 − ap, a, b. We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums P t∈Fp ap, t, b, P t∈Fp ap, a, t, Pp−1 t=0 a 2 p, t, b, Pp−1 t=0 a 2 p, a, t and Pp−1 …

Neutrality And Many-Valued Logics, Florentin Smarandache, Andrew Schumann

Branch Mathematics and Statistics Faculty and Staff Publications

ThisbookwrittenbyA. Schumann &F. Smarandache isdevotedtoadvances of non-Archimedean multiple-validity idea and its applications to logical reasoning. Leibnitz was the first who proposed Archimedes’ axiom to be rejected. He postulated infinitesimals (infinitely small numbers) of the unit interval [0,1] which are larger than zero, but smaller than each positive real number. Robinson applied this idea into modern mathematics in [117] and developed so-called non-standard analysis. In the framework of non-standard analysis there were obtained many interesting results examined in [37], [38], [74], [117].

There exists also a different version of mathematical analysis in that Archimedes’ axiom is rejected, namely, p-adic analysis (e.g., …

Auxiliary Information And A Priori Values In Construction Of Improved Estimators, Florentin Smarandache, Rajesh Singh, Pankaj Chauhan, Nirmala Sawan

Branch Mathematics and Statistics Faculty and Staff Publications

This volume is a collection of six papers on the use of auxiliary information and a priori values in construction of improved estimators. The work included here will be of immense application for researchers and students who employ auxiliary information in any form. Below we discuss each paper: 1. Ratio estimators in simple random sampling using information on auxiliary attribute. Prior knowledge about population mean along with coefficient of variation of the population of an auxiliary variable is known to be very useful particularly when the ratio, product and regression estimators are used for estimation of population mean of a …

Two Number-Theoretic Problems That Illustrate The Power And Limitations Of Randomness, Andrew Shallue

Andrew Shallue

This thesis contains work on two problems in algorithmic number theory. The first problem is to give an algorithm that constructs a rational point on an elliptic curve over a finite field. A fast and easy randomized algorithm has existed for some time. We prove that in the case where the finite field has characteristic 2, there is a deterministic algorithm with the same asymptotic running time as the existing randomized algorithm.