First, a little bit of fun history.
Somewhere around 250 AD, lived a Greek mathematician named Diophantus. His most well-known contributions to mathematics come from his 13 volume work Arithmetica in which he wrote and solved 189 mathematical problems and solved them with symbolic algebra.
There is not much known about Diophantus's life but there is a little math problem written about him (though it may be inaccurate) [1]. Can you solve it?
“‘Here lies Diophantus,’ the wonder behold. Through art algebraic, the stone tells how old: ‘God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father’s life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.”
I study Diophantine equations, which are polynomial equations in one or more variable with rational or integer coefficients. An example of such an equation is the following
where x, y, and z are positive whole numbers. This equation has no solutions and is straightforward to solve. Can you see why?
[Here's a hint; think of how 2 and 4 are different from 3.]
The goal in solving Diophantine equations is to determine if there are any solutions and if there are, to find all solutions. Most Diophantine equations require advanced techniques and tools to solve, such as Lehmer numbers, Diophantine approximations, elliptic curves in the modular approach, and linear forms in logarithms, among many other tools. Linear forms in logarithms are particularly useful in solving a special kind of Diophantine equation called Thue equations. Named for Axel Thue, Thue equations are homogeneous irreducible polynomials of degree three or greater.
Details of the families of Diophantine equations that I have worked with, some idea of the Thue equations that I am working on currently, and plans for my future work as well as work with students can be found in my Research Statement.
Opportunities for Undergraduate Research
I led a research group with SMALL in Summer 2021 and an REU group for AIM Up. They were even more productive than I could have hoped for. My talented research students at worked on several projects with me during the research class Spring 2020 online and continue on to the present.
Goals of these research projects include some or all of the following:
Community. Build a mathematical community with your peers, mentors, and mathematicians in the Number Theory network.
Respect. Foster respect, kindness, and awareness of your own and your peers’ needs.
Discover. Find the resources available for mathematical research, this includes uncovering where to search, what to look for, and how to assess the validity of sources.
Explore. Develop the skills to pick out the main purpose of a published mathematical paper, its method of proof, and the tools used to employ that method.
Aquire. Deeply study specific techniques for solving Diophantine equations through independent work with a small amount of guidance to help work around roadblocks.
Compute. Use Sage and/or Python to write short programs to perform computations to create examples and justify your hypothesis.
Proof. Apply learned methods to current open problems in Diophantine equations and together prove subsequent results.
Compile. Write up frequent reports on definitions, previous results, techniques applied to results.
A few project descriptions are listed here:
Survey of Topics in Diophantine Equations: In order to better understand the field of Diophantine equations, students are seeking out and complying what is currently known. Starting with a description of Diophantine equations and a brief history, this work will include names, descriptions, and recent work on various types of equations, as well as current unsolved problems in the field.
Cryptography Interacts: Cryptography is a deep and complex branch of mathematics that permeates many aspects of everyday life. To develop programming skills and a basic understanding of the historical cryptosystems, students will write and update the Sage Interact page for Cryptography. This is a public page that can be used by other students across the globe to learn and practice their programming and cryptography skills.
Extensions of Lebesgue-Nagell equations: We improve on current results on types of Lebesgue-Nagell equations based on student interest. One of the main tools for solving these equations uses the results of Bilu, Hanrot, and Voutier on defective Lucas and Lehmer numbers.
Continued Fractions over Non-Euclidean Fields: Daniel E. Martin’s work on extending continued fractions has a wide reach of applications including to Diophantine equations. By working through Martin’s work and applying it to families of relative Thue equations, a student aspiring to get a doctoral degree, will gain experience in a brand new area as well as study a valuable model of what their own future dissertation.
CUrrent & FOrthcoming Research Publications
B. Faye, E. Goedhart, and D. Wisniewski, “Families of Thue inequalities over imaginary quadratic number fields”, in preparation.
E. Goedhart, Y. Gurtas, and P. Harris, “A tree approach to the happy function,” in preparation.
E. Goedhart, B. Ha, L. McBeath, and L. Velasco, “On the Diophantine equations of the form λ1 U1 +λ2 U2 +...+λk Uk = wp^z1p^z2 ···p^zs ," in preparation.
M. Brucal-Hallare, E. Goedhart, R.M. Riley, V. Sharma, and B. Thompson, “Solving Quadratic and Cubic Diophantine Equations using 2-adic Valuation Trees,” submitted.
E. Goedhart, O. Kihel, and J. Larone, “Prime Rational Functions Over a Field,” submitted.
J. Carson, E. Goedhart, and P. Harris, “Sequences of consecutive factoradic happy”, Rocky Mountain J. of Math. 50 (2020), no. 4, 1241-1252.
E. Goedhart and H. G. Grundman, “On the Diophantine equation X^(2N) +2^(2α) 5^(2β) Y^(2γ) = Z^5”, Period. Math. Hung. 75 (2017), no. 2, 196–200.
E. Goedhart, “The Nonexistence of solutions to certain families of Diophantine equations”, Ph.D. dissertation, Bryn Mawr College: ProQuest/UMI, 2015.
E. Goedhart and H. G. Grundman, “Diophantine approximation and the equation (a^2 cx^k − 1)(b^2 cy^k − 1) = (abcz^k − 1)^2”, J. Number Theory, 154 (2015), 74-81.
E. Goedhart and H. G. Grundman, “On the Diophantine equation NX^2 + 2^L 3^M = Y^N ”, J. Number Theory 141 (2014), 214-224.
K. Berenhaut, E. Goedhart, and S. Stević, “Explicit bounds for third-order difference equations”, ANZIAM J. 47 (2006), no. 3, 359-366.
K. Berenhaut and E. Goedhart, “Second-order linear recurrences with restricted coefficients and the constant (1/3)^(1/3)”, Math. Inequal. Appl. 9 (2006), no. 3, 445-452.
K. Berenhaut and E. Goedhart, “Explicit bounds for second-order difference equations and a solution to a question of Stević”, J. Math. Anal. Appl. 305 (2005), no. 1, 1-10.
References:
[1] Pappas, T. “Diophantus’ Riddle.” The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 123 and 232, 1989.