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?

 
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‘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.
— Metrodorus

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

 
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where x, y, and z are positive integers. This equation has no solutions and is rather 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 homogenous 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.

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Publications

  1. E. Goedhart and H. G. Grundman, “On the Diophantine equation X^2N +2^2α5^2βY^2γ = Z^5”, Period. Math. Hung. 75 (2017), no. 2, 196–200.

  2. E. Goedhart, “The Nonexistence of Solutions to Certain Families of Diophantine Equations”, Ph.D. dissertation, Bryn Mawr College: ProQuest/UMI, 2015.

  3. E. Goedhart and H. G. Grundman, “Diophantine approximation and the equation (a^2cx^k − 1)(b^2cy^k − 1) = (abcz^k − 1)^2”, J. Number Theory, 154 (2015), 74–81.

  4. E. Goedhart and H. G. Grundman, “On the Diophantine equation NX^2 + 2^L3^M = Y^N ”, J. Number Theory 141 (2014), 214–224.

  5. K. Berenhaut, E. Goedhart, and S. Stević, “Explicit bounds for third-order difference equations”, ANZIAM J. 47 (2006), no. 3, 359–366.

  6. 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.

  7. 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.