Note: The versions of these papers that I have put here are preprints. They may differ from the final published version.

In Preparation

[29] (with H. Godinho and B. Miranda) Sextic forms over \( \mathbb{Q}_2(\sqrt{5})\).

Submitted

Abstract: Let \( a \) be an integer greater than 2, and define a sequence recursively by \( G_0 = a\), \( G_1 = 1\), and \( G_n = G_{n-1} + G_{n-2} \) for all integers \( n \). Let \( G(x) \) be the generating function for this sequence. In this article, we give a method to find all rational values of \( x \) such that \( G(x) \) is an integer, and give several families of explicit solutions. For many values of the parameter \( a \), these solutions are the only ones.

Accepted for Publication / In Print

Abstract: In this article, we study equations of the form
\[ a_1 x_1^k + \cdots + a_s x_s^k + b_1 y_1^n + \cdots + b_s y_s^n = 0, \] where all the coefficients are integers. The main result of the article is to give a sharp upper bound on the smallest value of \( s \) (as a function of \( k \) and \(n\) ) that guarantees that this equation has nontrivial solutions in the \( p\)-adic field \( \mathbb{Q}_p \) for all primes \( p\). We also give the exact smallest value of \( s \) for various choices of \( k\) and \( n\).

Abstract: Suppose that \(\mathbb{K}\) is any quadratic extension of the \(p\)-adic field \(\mathbb{Q}_2\). We show that every diagonal form of degree \(d\) with coefficients in \(\mathbb{K}\) having at least \(d^2 + 1\) variables has a nontrivial zero in \(\mathbb{K}\).

Abstract: The Mersenne numbers are numbers of the form \(M_n = 2^n - 1\), where \(n\) is a positive integer. The main theorem in this article is to give a description of all the Mersenne numbers which have at most 3 distinct prime factors. We also show that it is rare for a Mersenne number to have few prime factors, and additionally study the equation \(M_m + M_n = 2p^a\), where \(p\) is a prime number.

Abstract: This is a sequel to my paper [13] below. In this paper, we find the value of \(\Gamma^*(k)\) for all values of \(k\) up to \(k = 64\). We accomplish this through essentially the same methods as in [13], although some of our computational algorithms are improved, and we are able to treat some situations theoretically that I treated computationally in [13].

Abstract: The number \(\Gamma^*(k)\) is defined as the least integer \(s\) such that the diagonal form \[ a_1 x_1^k + \cdots + a_s x_s^k\] is guaranteed to have a nontrivial zero in every \(p\)-adic field \(\mathbb{Q}_p\), regardless of the coefficients. An old conjecture of Norton posited that \(\Gamma^*(k) \equiv 1 \pmod{k}\) for all \(k\). Although this conjecture has long been known to be false, only 3 counterexamples have been produced, with two of them only coming in recent years. In this article, we produce infinitely many values of \(k\) for which Norton's conjecture is false.

Abstract: In this article, we answer the following question about graph pebbling. Consider a cycle graph with \(n\) vertices and all edges directed counterclockwise, and choose a positive integer \(d\) with \(d < n\). We determine the smallest number \(s\) such that if \(s\) pebbles are placed on the graph, then there is at least one pebble which can be moved a distance of at least \(d\) vertices away from its original position through a sequence of pebbling moves.

Abstract: In this article, we calculate an upper bound for the value of \(\Gamma^*(k,p)\), which gives the smallest number of variables necessary to guarantee that any additive form of degree \(k\) with integer coefficients has a \(p\)-adic zero, and also give a condition under which this upper bound is actually an equality. Additionally, we calculate the value of \(\Gamma^*(54)\), which is the smallest number of variables needed to guarantee that any additive form of degree 54 with integer coefficients has a zero in every \(p\)-adic field \(\mathbb{Q}_p\).

Abstract: Consider an additive form \(a_1 x_1^6 + \cdots + a_s x_s^6\) whose coefficients are integers of one of the six unramified quadratic extensions of the \(p\)-adic field \(\mathbb{Q}_2\). In this article, we show that if the form has at least 9 variables, then it must have a nontrivial zero. For four of these extensions, we show that no smaller number of variables has this property.

Abstract: Consider an additive form \(a_1 x_1^d + \cdots + a_s x_s^d\) whose coefficients are all 2-adic integers. In this article we give an exact formula, in terms of \(d\), for the smallest number of variables which guarantees that this form has a nontrivial zero in the 2-adic integers regardless of the values of the coefficients.

Abstract: Major League Baseball's decision more than a decade ago to add wild card teams to its playoff structure remains controversial even now. In this article, we attempt to support the notion that there should be wild card teams. To do this, we make a very simple probabalistic model of Major League Baseball's final regular-season standings. Then we use this model to predict how often the team that becomes the wild card will have a better record than the worst division-winning team, and calculate the expected rank in the league of the worst division winner. Perhaps surprisingly, our model predicts that the worst division winner should finish 7th or worse in the league more than 12% of the time.

Abstract: In this article, we treat sequences formed by the same procedure as the ones in the article [14] below. We show that the patterns described in that article will always exist whenever \(1 < d < n-7\). This majority of this work was done by Alyson Fox while she was my student in the Hauber summer research program at Loyola.

Abstract: This article shows that a system of two homogeneous additive polynomials of degree 6 with integer coefficients will have a nontrivial \(p\)-adic solution for any prime \(p\) provided only that the system has a total of 73 variables.

Abstract: Suppose that \(k\) and \(n\) are odd integers. We show that any system of diagonal forms of degrees \(k\) and \(n\), with integer coefficients, in at least \(k^2 + n^2 + 1\) variables has a nontrivial zero in the field \(\mathbb{Q}_p\) for each prime \(p\).

Abstract: Consider a sequence defined recursively by the following procedure. Pick numbers \(n\) and \(d\), and set \(a_1 = n\) and \(a_2 = n+d\). Then for each \(k > 2\), set \(a_k\) to be the smallest number greater than \(a_{k-1}\) which cannot be written as a sum of previous terms of the sequence. Experimentation shows that the terms of the sequence resulting from this procedure appear to have several very interesting patterns. We show that these patterns really exist in the cases where \(n = 1\), or where \(d = 1\), or where \(n = d\). The majority of this work was done by Michael Paul while he was my student in the Hauber summer research program at Loyola.

Abstract: The function \(\Gamma^*(k)\) gives the smallest number of variables required to ensure that an additive form of degree \(k\) has a nontrivial solution in \(p\)-adic integers for every prime \(p\). In this article, we evaluate \(\Gamma^*(k)\) exactly when \(k = 14\), 20, 24, 26, 27, 29 and 31. With these results, the exact value of \(\Gamma^*(k)\) is now known whenever \(k < 32\).

Abstract: Suppose that \(a\) is a \(p\)-adic number. In this paper, we consider the problem of calculating the \(p\)-adic digits of \(\sqrt{a}\). We do this using two fixed-point methods with quadratic and cubic rates of convergence. For each method, we calculate the minimum number of correct digits obtained after \(n\) iterations, and the number of iterations required to obtain an approximation of any given accuracy.

Abstract: In this expository paper, we show how rootfinding methods from numerical analysis (Newton's method and the secant method) can be used to calculate inverses of numbers modulo powers of primes. In undergraduate courses, the fields of numerical analysis and number theory can appear to be unrelated, and we believe that students may enjoy seeing a connection between them.

Abstract: In this expository paper, we derive a formula for the sum of a series of sines or cosines, where the angles form an arithmetic progression. This derivation of the formula was originally given in the journal Arbelos, which is now sadly out of print. The derivation is interesting because it does not make any use of complex numbers.

Abstract: This paper is similar to [7] below, except that we are able to remove the restriction that the degrees of the forms must be all different. Given a system of diagonal forms over \(\mathbb{Q}_p\), we ask how many variables are required to guarantee that the system has a nontrivial zero. We show that if the prime p satisfies \[ p > \mbox{(largest degree)} - \mbox{(smallest degree)} + 1, \] then there is a bound on the sufficient number of variables which is a polynomial in the degrees of the forms. A result of Lewis & Montgomery implies that any bound which works for all primes must exhibit exponential growth. So this result can be thought of as bounding how small \(p\) can be before exponential growth is required.

Abstract: In 1966, Davenport & Lewis published their paper Notes on congruences III, in which they proved that under some mild conditions, a system of two additive forms must have a nonsingular simultaneous zero modulo any prime number. In their paper, they ask whether the theorem is true in general finite fields, and point out that one of their key lemmas is no longer true in this situation. In this paper we answer their question in the affirmative, proving that under the same conditions, a system of two additive forms over a finite field must have a nonsingular simultaneous zero. We then apply this result to obtain an upper bound on the number of variables required to ensure that a system of two additive forms of equal degrees has a nontrivial zero in a \(p\)-adic field.

Abstract: It is known that any system of diagonal forms has a nontrivial \(p\)-adic zero for all primes \(p\) provided that the number of variables is large enough in terms of the degrees. A result of Lewis & Montgomery shows that the required number of variables must exhibit exponential growth. However, a theorem of Ax & Kochen states that if \(p\) is large enough, then a bound which exhibits only polynomial growth suffices. The purpose of this paper is to bound the primes for which exponential growth is required in the situation where the degrees are all different. In particular, if the degrees of the forms are \(k_1 > \cdots > k_R\) and if \(p > k_1 - k_R + 1\), then we give a bound for the required number of variables which has only polynomial growth. This bound is larger than the Ax-Kochen bound, but we note that it applies for primes which are smaller than the largest of the degrees.

Abstract: In this paper we develop a bound on the number of variables required to guarantee that two diagonal homogeneous polynomials of different degrees \(k\) and \(n\) with coefficients in \(\mathbb{Q}_p\) have a nontrivial simultaneous zero in \(p\)-adic integers. If \(k\) and \(n\) are large and close together, then this is the best known bound. We also prove a much stronger bound in the case when at least one of \(k\) and \(n\) is odd.

Abstract: A recent paper in Math Magazine gave a proof that there are exactly \(p^2(p^2 + 2p - 1)/2\) two-by-two matrices with entries in \(\mathbb{Z}/p\mathbb{Z}\) and both eigenvalues also in this field. The proof appeals to a theorem from abstract algebra in the key step. In this note, we give a simpler proof that does not require abstract algebra, using only concepts which could be taught in an undergraduate number theory course.

Abstract: It is known by work of Ax & Kochen that given any natural number \(d\), any homogeneous polynomial of degree \(d\) in \(d^2 + 1\) variables with coefficients in \(\mathbb{Q}_p\) has a nontrivial zero provided that \(p\) is sufficiently large. In this paper, we give upper bounds on how large \(p\) needs to be when we have either \(d = 7\) or \(d = 11\).

Abstract: The main result of this paper is that any system of \(R\) diagonal (additive) homogeneous polynomials of degree \(k\) in at least \(4 R^2 k^2\) variables with coefficients in \(\mathbb{Q}_p\) has a nontrivial simultaneous zero in \(p\)-adic integers. This improves on work of Brudern & Godinho, and is the best known bound when \(k\) is even and suitably large in comparison to \(R\). A version of this theorem is also developed for the situation in which \(\mathbb{Q}_p\) is replaced by a finite extension of \(\mathbb{Q}_p\).

Abstract: In this expository paper, we introduce the reader to the analysis of musical chords using the additive group \(\mathbb{Z}/12\mathbb{Z}\) and the twelve-tone operators, which can be thought of as functions on \(\mathbb{Z}/12\mathbb{Z}\).

Ph.D. Thesis

[1] Forms in many variables over p-adic fields. Ph.D. thesis, University of Michigan, 2000. [PDF] [PS] [DVI]

Abstract: This is my Ph.D. thesis. The content includes that of papers [3] and [4] above, and also some of [7]. Additionally, we elaborate on work of Skinner to give a bound on the number of variables necessary to guarantee that there exists a nontrivial simultaneous zero of a system of \(R\) homogeneous additive polynomials of degree \(p^t\) with coefficients in a finite extension of \(\mathbb{Q}_p\). This bound does not depend on the degree of the field extension. There is also an introduction to the field of study of zeros of homogeneous equations over \(p\)-adic fields.