What follows is a list of some of the projects that faculty members in the department of mathematics have suggested as suitable for undergraduate research projects. Some of the projects that are listed were suggested some time ago, but faculty typically are still willing to work with students on these or a related project.
Students who wish to participate can register and receive credit for an independent study, or, may be able to obtain URA funding to get paid to work on these projects.
Details of the project requirements will be worked out between the faculty supervisor and the student. Some of these projects require little background and are suitable for freshmen or sophomores, while others require knowledge of linear algebra, ordinary differential equations, or group theory. This list is by no means exclusive: any student with a particular interest in some area of research is encouraged to seek out a faculty supervisor, or to contact the URA Program Coordinator for help on finding a suitable faculty member.
Advisor: Jim Cushing
Prerequisites: A first course on differential equations & some familiarity with matrices. Some proficiency with a programming language would be useful (MatLab, Maple, Mathematica, etc.).
Summary: A fundamental goal in ecology is to gain an understanding of how ecosystems are built and maintained. In this regard, it is important to understand how several species can coexist while competing for limited resources or otherwise interfering with each other’s well being. To attain diversity within an ecosystem numerous species must find ways to coexist while in ecological competition. The life cycle strategy adopted by individuals (the timing of developmental stages, when and how often to reproduce, and so on) is an important component that bears on the ultimate survival or extinction of a species. To give but one example: why do some species of cicadas adopt a life cycle in which adults emerge only once in every 17 years? How is this a successful strategy? Why do other species not adopt such a strategy, but instead overlap their generations? Mathematical models for the dynamics of competing species that include a description of the life cycles of individuals can help determine which strategies result in increased chances for coexistence and which lead to probable extinction.
Students in this project will have the opportunity to study and derive models of competition (and/or predation, parasitism, etc.), to analyze them mathematically, and/or to carry out simulation studies on computers.
Advisor: Chris Bergevin
Prerequisites: Math 254 or 355, Some experience with Computations (e.g. Matlab, Java or C..) is helpful
Summary: The ear acts as a
biological microphone. Its most basic function is to sense incoming
acoustic stimuli from the world around us and convert that mechanical
information into electrical signals which travel via the auditory nerve
to the central nervous system. Thus, it might seem a bit surprising
that a healthy ear actually produces and emits sound (which can be
measured via a microphone in the ear canal). These sounds are called
otoacoustic emissions (OAEs) and can arise either spontaneously or by
an evoking external stimulus. These emissions are generated in the
inner ear (cochlea) and are presumably a result of the mechano-electro
and electro-mechano transduction mechanisms responsible for converting
sound into neural impulses. Since the cochlea is encased in very hard
bone and the sensory structures are highly sensitive (i.e. easily
subject to damage), direct physiological observation of the inner ear
is difficult. However, OAEs provide a non-invasive window into the
inner ear and allow us to better understand the underlying physiology.
One of the striking features of evoked OAEs is there non-monotonic
dependence upon the input stimulus intensity. The basis for this
nonlinear behavior is currently poorly understood. This project will
focus on elucidating the physiological mechanisms underlying the
non-monotonic behavior via a computational approach. An initial step
will be to look at predictions of various proposed models and compare
those to real OAE data. This project is well suited for students who
are interested in applied math and its applications in biology/sensory
physiology. While a background in biology is not necessary, some
experience with computation (e.g. basic coding in Matlab) will be
helpful.
Advisor: Nick Rodgers
Prerequisites: Understanding of some basic algebraic structures like groups, rings, and fields, at the level of Math 415A.
Summary: An elliptic curve is the set of points satisfying a cubic equation, for example y^2 = x^3 + ax + b. The study of elliptic curves is remarkably rich, using techniques from algebra, geometry, and complex analysis. The scope of applications is nearly as broad, ranging from the solution of centuries-old problems in Number Theory (including, most famously, Fermat's Last Theorem) to cutting-edge applications in Cryptography.
The rich algebraic structure of elliptic curves originates with the "chord-tangent composition law," which is a geometric way to make the rational points on the elliptic curve into a commutative group. A classical result in the field states that this group is finitely generated, so that beginning with a finite set of points, one can obtain all of the (possibly infinitely many) rational points on the curve via repeated application of the chord-tangent law. The size of this generating set, loosely speaking called the rank of the elliptic curve, is a little-understood quantity that is often difficult to compute in practice. The rank of a "random" elliptic curve is typically small, but nevertheless a standard conjecture states that there are elliptic curves with arbitrarily large rank.
The goal of this project is to introduce the basic properties of elliptic curves, and to investigate some more precise conjectures related to how the rank behaves within certain families of elliptic curves.
Advisor: James Cossey
Prerequisites: Linear algebra, some familiarity with groups would help but is not required.
Summary:The braid group B(n) can be thought of as n strings hanging side by side from a fixed pole, and the elements of the braid group are simply the "braids" that are possible, i.e. all the possible crossings of the strings. These groups admit a remarkably rich structure, and we will study the structure using ideas from group theory.
The braid groups can also be used to create crypto-systems, where breaking the code amounts to answering basic questions in the braid group, such as the conjugacy or root problems. We will investigate these problems and attempt to make and/or break good braid cryptosystems.
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Advisor: James Cossey
Prerequisites: Linear algebra, some familiarity with groups would help but is not required.
Summary:Error-correcting codes are an extremely useful method of transmitting large amounts of information through a noisy channel; for instance cell-phones and satellites use these methods. Recent developments in the theory of "space-time" codes seem especially promising.
In this project we will study recent progress that has been made in applying basic ideas from finite group theory to the development of space-time codes. We will then attempt to use these "fixed-point" methods to generalize some of these results.
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Advisor: Marty Greenlee
Prerequisites: Second semester calculus (Math 129?), and the willingness to do illustrative calculations, possibly with a calculator, but better with a computer.
Summary:The error formula formulas for common quadrature rules of numerical integration are given in terms of derivatives of the function whose integral is being approximated, e. g.,
What happens when those derivatives are questionable or nonexistent, as e. g. with graphical data? An integration by parts formula for step functions (a special case of integration by parts in Stieltjes integrals) can be used to estimate the convergence rate with one bounded derivative, or even with integrable singularities. This information does not seem to be available in the literature, at least in an organized and understandable fashion. The project is to examine the utility of this approach.
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Advisor: Marty Greenlee
Prerequisites:Linear algebra (Math 215), some familiarity with convergence/divergence of sequences, and willingness to do some calculations.
Summary: The iterative methods known as the power method, inverse power method, and Rayleigh quotient method can be used to find eigenvalues of matrices, and are basic to the QR factorization algorithm for matrices. These iterative methods can be recast as fixed point theorems for contraction maps via successive approximations. This approach differs considerably from the literature on numerical linear algebra. The project is to explore this different point of view, and seek new conclusions.
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Advisor: Ksenija Simic
Prerequisites: There are no formal prerequisites, but students should be familiar with formal mathematical reasoning, the kind used in MATH 215 (Linear Algebra) or MATH 323 (Formal Mathematical Reasoning and Writing).
Summary:
As most students get little (if any) exposure to mathematical logic in the course of their studies, my goal for this project is to introduce the interested student to mathematical logic in an engaging and non-intimidating manner. The project topic is loosely defined, as it will depend on student backgrounds and interests. If a student has no previous experience with mathematical logic, then we will spend the first part of the semester getting acquainted with the subject (propositional logic, first-order logic, relevance to mathematics, historical developments, some seminal results such as the completeness theorem and the incompleteness theorems).
Once the student is ready to choose a topic, we will find an appropriate one together. I am mainly interest is in intuitionistic logic (description follows below), and its use in proving the usual theorems of mathematics, but would also be willing to work on a project in other areas of logic, especially category theory. I am also interested in working with students with backgrounds in linguistics or philosophy or those interested in the history of mathematics, in which case we could study a topic relevant to all fields involved.
Description of intuitionistic logic
Intuitionistic (or constructive) logic rejects the claim that a statement that is not false is true (or equivalently, that every statement is either true or false), a claim that we take for granted in mathematics, though not necessarily in everyday reasoning. Because of this restriction, a constructive notion of proof is much more rigorous and a constructive proof gives more computational information than a classical one, not to mention that the methods and results are highly unusual. Many results proved classically do not apply constructively. Intuitionistic logic has many applications in theoretical computer science.
if a student chooses to do this project, he/she would first learn about the basic notions of proof theory and model theory for classical first-order logic and then explore the differences between classical and intuitionistic logic, find examples of facts that hold in one, but not the other, using tools of mathematical logic such as formal systems and model theory. We could also look at familiar theorems of mathematics and find examples of ones that can be proved constructively.
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Advisor: Arlo Caine
Prerequisites: Linear Algebra, Abstract Algebra, Real Analysis, and curiosity.
Summary: The tilings of the plane that many of us are familiar with are periodic, such as a tiling of unit squares for example. This tiling is periodic the sense that a translation vertically or horizontally by one unit doesn't change the image.
Penrose Tilings are a class of aperiodic tilings of the plane; they admit no translational symmetries. It turns out that the set of all Penrose Tilings makes up a very strange space with many fascinating properties involving, among other things, the fibonacci numbers, and the golden ratio! (Commutative) Geometry, the study of spaces, ignores all of these properties, but one can use the finer tools of non-commutative geometry to explore this amazing space.
In this project we will investigate some of the marvelous properties of Penrose Tilings and study their non-commutative geometry. To do so we will, along the way, survey ideas from Dynamical Systems, Topology, Analysis, and Algebra.
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Advisor: Doug Pickrell
Prerequisites: A good grasp of vector calculus and linear algebra.
Summary: Given a polygon in the plane, one can compute the area by cutting the polygon into triangles and rearranging them into a rectangle. Thus every polygon in the plane is scissors congruent to a rectangle of equal area. In the 1840's Gauss asked if an analogous statement was true for a 3 dimensional simplex (i.e. a solid with planar faces) in 3-space. In 1900 Dehn proved that the answer is “no”, by showing that another number (Dehn invariant, the sum of the product of the lengths of edges and corresponding dihedral angles), in addition to volume, is preserved under scissor congruence. Sixty years later Jensen proved that two 3-simplices are scissors congruent if and only if they have the same volume and Dehn invariant. In turn Jensen asked if the analogous statement was true for simplices in spheres and other non-Euclidean geometries. Many of these problems are unresolved.
In this project we will investigate these and related issues involving volume of simplices in Euclidean and spherical 3-space.
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Advisor: Robert Indik
Prerequisites: Linear algebra, and strong familiarity with complex number. Computer experience helpful.
Summary: Light such as that that comes from lasers can be described in terms of intesity and phase. When images are captured, they record the intesity of the light. If the phase can be found as well, it is possible to recover full information about the light, and to undo the effects of poor focusing or of blurring due to turbulence in the air. I am investigating the practicality of reconstructing phase information from images taken in two different focal planes (the near filed and the far field).
This problem can be recast into the question: Given the magnitude of a complex function f(x,y), and the magnitude of its Fourier transform, can one reconstruct the original function?
If one specializes to the case of discrete functions and images such as can be captured electronically, the question becomes: Given the magitudes of the complex entries in an N by N array, and the magnitudes of the discrete Fourier transform of that array, can one reconstruct the complex array?
Students participating in this project will explore these questions initially in very small test cases, and test out potential algorithms for efficiently solving for phase.
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Advisors: Donald Myers (Mathematics), Robin Harris (College of Public Health, Program in Epidemiology), Mary Kay O'Rourke (Division of Community and Environment), and Séumas Rogan (Epidemiology Program). (Please contact Donald Myers to discuss this project.)
Prerequisites: Linear algebra, computer experience (word processing, spreadsheet, statistical software or willingness to learn, GIS or willingness to learn), probability theory and statistics desirable.
Summary: The Atlas of Cancer Mortality in the United States: 1950-1994 tabulates the distribution of cancer in the United States by county. The creation and utility of the atlas requires adoption of various assumptions. In particular, it implies that the cancer risk is equal across a county, and that the reported cancer risk represents the cancer risk for all residents of that county. If the county is fairly homogeneous in both the characteristics of the underlying population and exposure risks, then this assumption may be reasonable. However, for many Western U.S. states, this assumption may be inappropriate. These states often are divided into only a few counties, each covering large geographic spaces and having uneven distributions of populations. These characteristics make it unlikely that county-level statistics fairly represent the range of actual county experiences.
The overall goal of this research project, is to examine the geographic variation in the association between cancer risk and arsenic. Arsenic exposure may be a causal agent in the development of bladder, lung, kidney, and skin cancers. Furthermore, arsenic is known to vary across geographic locations. GIS technology has made it more feasible to link multiple sources of descriptive attribute information for various geographic levels with health outcome data.
Several geographically delineated data sets exist in Arizona that allow for exploration of the relationship between arsenic exposure and cancer occurrence. Geocoded cancer incidence and mortality data are available from the Arizona Cancer Registry for bladder, kidney, and lung cancers. Skin cancer data are available from a completed population-based case-control study. Arsenic concentrations are available from a multimedia, multipathway survey conducted in Arizona.
The researchers will examine cumulative, aggregate, and cumulative-aggregate arsenic exposures, as potential doses, with the incidence of specific cancers. They will determine the homogeneity of the associations across various geographic scales.
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Advisor: Donald Myers
Prerequisites: Linear algebra, computer experience (word processing, graphing software, R software or willingness to learn, GIS or willingness to learn, FORTRAN or C programming useful but not mandatory), probability theory and statistics desirable but not mandatory.
Summary: NDVI data is widely used as an index of vegetation. To detect whether change has occurred in vegetation then it is necessary to detect whether change has occurred in a time sequence of NDVI data files. Because change, if it occurs, may not occur in all parts of a region, i.e., for all pixels in the data sets, some form of statistic is needed that will summarize how the pixel values relate to each other spatially and also temporally. A space-time variogram is one such tool. Then it is necessary to estimate and fit a valid space-time variogram to the data. In this study, data for the years 1990-1999 from the Oregon Pilot Study will be used. The product-sum model variogram not only provides a method for constructing valid space-time variograms but also a way to estimate and fit the model. The characteristics of the fitted model will be interpreted as a tool to ascertain whether there has been a change in the vegetation.
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Advisors: Dan Madden, William McCallum.
Prerequisites: None.
Summary: Attached to a polynomial with integer coefficients is a certain number, called the Mahler measure, which is computed in terms of the roots of the polynomial. There are also other ways to define this measure. Mahler measure is not very well understood, and the idea of this project is to explore it by re-establishing some known results using new techniques. To learn a little more, see our detailed project description.
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Advisors: Richard Thompson (Mathematics) and Dennis Doxtater (Architecture, Planning, and Landscape Architecture)
Prerequisites: Knowledge of trigonometry, willingness to learn spherical trigonometry and some statistics.
Summary: This is a project outside the math department. Dr. Thompson has helped Dr. Doxtater get started with a project in archeology and an application using the MATHCAD program. A piece of a help wanted description from Dr. Doxtater is included below. Interested students should start by contacting Dr. Thompson.
The effort is to determine whether existing, accurate large-scale geometric patterns between certain archaeological sites and significant natural features are purely coincidental or were designed. I need someone to build a computer application that can run a large number of tests where the location of archaeological sites are randomly varied in relation to fixed natural features. The work can either be for independent academic credit without compensation or funded from grant sources. While the actual test will focus on existing patterns between Anasazi sites in Chaco Canyon and distant mountains and other natural features, the software needs to be generic and usable for other cultural landscapes.
In the Chaco case, there are five distinct geometric patterns that involve both natural and built sites. These five patterns are linked in one way or another through common sites or geometric features. They form a constellation with a seemingly logical sequence of linkage and development. In each of the five patterns, it is possible to state the geometric relationship of the archaeological site to natural features as an angular deviation (degrees). Thus the accuracy of the overall constellation can be defined as the average of the five deviations. The strategy of this introductory research is to calculate the average deviation of randomly located archaeological sites for the particular patterns in a constellation. Conduct this test many times and compare to the average deviation of the existing constellation.
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Advisor: Tom Kennedy
Prerequisites: Basic probability concepts, such as the distribution function of a random variable. Familiarity with some topics from MATH 468, in particular Markov chains, would be useful, but is not essential. Since the project will consist of computer simulations, programming skill is needed. However, programs to simulate these self-avoiding walks can be surprisingly short.
Summary: To construct a random walk, each time you take a step you randomly choose one of the four directions north, east, south or west. In a self-avoiding random walk you are not allowed to visit the same place more than once. While very little has been proved about these walks, they can be simulated rather easily on a computer, and so can be studied numerically. Here are a couple of pictures of computer generated self avoiding walks:
If you run an ordinary random walk for a long time and then look at it from far away, it looks like a stochastic process called Brownian motion. Now suppose you do the same thing for self-avoiding random walks. Does it look like a stochastic process, and if so what can you say about the process? The goal of the project is to study this question empirically, i.e., by simulating lots of self-avoiding walks and seeing what we can say.
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Advisor: William McCallum
Prerequisites: Linear algebra and some facility with computers. A first course in abstract algebra (groups, rings, fields) would also be quite useful, as would more in-depth knowledge of computer science (such as complexity theory).
Summary: These projects were initially listed by professor Ulmer. Professor McCallum has taken over Cryptography (it is a friendly takeover).
Cryptography has probably been with us almost as long as organized military activity, but recently it has become much more important to the public at large. The widespread availability of networked computers and their increasing use in business makes it crucial to be able to communicate securely over unsecure channels. On the other hand, the same widespread availability of powerful computers makes it possible for junior high school students to break, by brute force, codes that would have stoppped the best minds of earlier generations.
One may crudely divide cryptographic systems (codes) into two classes: symmetric key and public key. With symmetric key codes, the parties communicating must share a secret (the key) in advance and the same key is used for encryption and decryption (hence the label symmetric). These codes typically run fast and are often used for commercial purposes such as bank transfers. Until quite recently, all cryptographic systems were symmetric key systems.
Public key cryptographic systems use different keys for encryption and decryption and, very importantly, knowing one of these keys tells you nothing about the other. Such systems make it possible to perform amazing feats like proving that you know a secret without revealing the secret, or establishing a secret with a stranger using open channels of communication, in such a way that an eavesdropper who hears the entire conversation will not be able to reconstruct the secret.
Here are three possible projects for student research:
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Advisor: Timothy Secomb
Prerequisites: Ordinary differential equations, linear algebra.
Summary: The microcirculation is an intricate network of tiny blood vessels that carries nutrients to every part of a living tissue. Current projects are aimed at developing mathematical models for several aspects of the system's function. For example, networks continually adjust their structures in response to local signals, during growth and when metabolic requirements change. This process can be modeled using a system of ordinary differential equations to describe changes in vessel diameters. The problem is to determine what types of responses can lead to stable, adequate network structures consistent with experimental observations.
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Advisor: David Lomen
Prerequisites: None.
Summary: Topics chosen by the student in areas of mathematical modeling dealing with fluid motion (soil physics, physiology, engineering), with differential equations (pharmacokinetics, biological, chemical, or physical processes, ...) or with different ways to teach mathematics or the development of innovative teaching materials.
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Advisor: Moysey Brio
Prerequisites: MATH 223, 215, and 355 useful.
Summary: Code development, grid generation and optimization of finite element and finite differencing with applications to computation of guided modes in photonic structures, pulse propagation dispersive, active and nonlinear media, phase locked mode propagation in semiconductor lasers.
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Advisor: Marek Rychlik
Prerequisites: Basic calculus sequence required, proficiency in a programming language required, Fourier analysis helpful, probability helpful, linear algebra helpful.
Summary: Study the motion of a billiard ball on a table of an arbitrary shape. Only elliptic tables are known to be integrable. Other tables are conjectured to be chaotic. In the course of this project the student would perform simple numerical experiments to measure the chaotic behavior of tables of various shapes.
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Advisor: Marek Rychlik
Prerequisites: Calculus and linear algebra necessary, Fourier analysis helpful.
Summary: The student would collect samples of speech, using a computer with a sound card and a microphone. Subsequently, these samples would be visualized, using software like MATLAB. The individual sounds, like the ones produced by saying “aaaah” or “oooh” produce patterns which are referred to as attractors. These patterns will be subjected to a mathematical analysis to explore the possibility of distinguishing between them.
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Advisor: Marek Rychlik
Prerequisites: Discrete math, programming language.
Summary: A neural network is a system that will produce “correct” responses to a range of inputs. The traditional method of training of neural networks is the back propagation algorithm and it is known to be very slow. The student will examine a new method of genetic breeding and compare it to the old method by performing numerical experiments.
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Advisor: Joseph Watkins
Prerequisites: Varied.
Summary: My recent work involves interaction between the mathematics and biology communities. Under the auspices of the Southwest Regional Institute in the Mathematical Sciences, we are developing ways to communicate research ideas in population biology. This is now being pursued on the biology of bees and soon we will start investigating questions on HIV both its immunology and its epidemiology. Undergraduate projects here come in two types - one is to work with scientists and teachers to develop strategies for communicating these ideas to broad community.
At this moment, we are just starting to write software on bee population models. These models will become tools for high school students, beekeepers and for bee scientists. These groups are concerned about the Africanization of the European honey bee population and mite infestations of bee hives. We hope that these tools will help understand the nature of these events and suggest methods for remediation.
The HIV models will also require some software development. The issues in the immunology are optimal drug protocols for improving the lives of those with HIV. The epidemiological issue is to investigate what changes in behavior will have the biggest impact in the spread of the virus.
Separated from my activities with the Regional Institute, I am also looking into issues involving enzyme kinetics. When the number of enzyme molecules and substrate molecules was high, mass action equations were an adequate mathematical model of the dynamics of the enzyme reactions. As biochemistry is learning more about enzymatic reaction in which the number of molecules is smaller, the stochastic or random effects are playing a more important role. Here we will be writing programs that find the importance of these effects and looking into the experimental data for verification.
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Advisor: Daniel Madden
Prerequisites: Pattern recognition skills and computer skills required. Number theory and familiarity with proof techniques helpful.
Summary: A continued fraction is a number written in the form:
The only numbers that have continued fractions that terminate are the
rational numbers. But all numbers, rational and irrational numbers,
have unique continued fraction expansions. Some irrational numbers
have continued fractions that consist of a repeating pattern of
numbers 
. We denote these by placing a
bar over the repeating part; for example:
These are exactly the irrational numbers that satisfy quadratic polynomials with integer coefficients.
The problem concentrates on the continued fraction expansions
of numbers of the form 
where n is an integer. These numbers have repeating
continued
fraction expansions. Some of these expansions can be expressed by a
formula that can easily be verified. For example:
Many such formulae are easy to notice in any good list of
expansions. The interesting formulae are those where the number of
terms in the repeating pattern grows larger as the number under the
square root grows larger. These formulae are harder to pick out and
more difficult to verify. The problem is to find, and perhaps verify,
formulae for 
where
the length of the
repeating pattern is also a function of n.
This problem requires a certain amount of computer skills, and
would require some initial study of continued fractions to get started.
The problem is related to a famous open conjectures of Gauss. For
example, if one found a formula (highly unlikely) for where f(x)
is a polynomial and where the
repeating pattern had length n, this alone would
probably prove
Gauss' conjecture.
You may be interested in seeing some of the work done by Justin Miller.
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