Research

A short version of my publication list is available on my CV. Here I expand on many of my papers, largely for my students who ask about my scholarship and research.

My academic research falls into three broad categories, which sometimes (happily) overlap, namely:

Computational and Elementary Analytic Number Theory

My Ph.D. was in this field; Elementary Number Theory and Algorithms. In particular, my work put an upper bound on the Brun's Constant -- the sum of the reciprocals of the twin primes. This involved, in part, creating algorithms to approximate this sum over various regions of the integers. If you'd like, you can read my thesis!

More generally, I'm interested in explicit results in Elementary Number Theory. This involves using a combination of theoretical and computational work. See for example my papers in this area:

  1. Families of values of the excedent function sigma(n)-2n (with R. Dean, R. Erdman, E. Lycette, M. Pidde, and D. Wheel). Missouri Journal of Mathematical Sciences, 27(1), pp. 37--46, (2015).

This paper arose out of my first Mathematics Honors Seminar, when my students built on work done by students Davis and Kraght the year before (see the paper on the values of s(n) - n below). Writing it was a wonderful experience, and the theorems came together via a true group effort by all of the students. I was (and am) so proud of them!

2. On formal graph-theoretic definitions of the Ihara zeta function (with C. Storm). Advances and Applications in Discrete Mathematics, 14(2). pp. 151 - 168 (2014).

Zeta functions are fascinating tools which take a large amount of information (concerning prime numbers, say), and capture all of it in one function. I've published two papers on zeta functions -- this one, concerning graphs, and another, concerning juggling sequences. This paper is both a survey of various work and definitions on the Ihara Zeta function (which captures information about "prime cycles" in a graph), and to offer another which seems to be more useful in some situations.

3. New Bounds and Computations on Prime-Indexed Primes (with J. Bayless and T. Olveira e Silva). Integers, 07/2013; 13: #A43, pp. 1-21, (2013).

Imagine listing all the prime numbers: 2, 3, 5, 7, 11, ... . We are interested, in this paper, in the primes whose position on that list is prime. For example, we want the 2nd, 3rd, 5th, 7th prime, etc. The are the Prime-Indexed-Primes What can we say about these numbers? How accurately can we predict how many there will be up to any point x? The paper addresses these, and many other, questions concerning the Prime-Indexed Primes. We also conjecture that every even number greater than 80612 is the sum of two Prime-Indexed Primes.

4. On the values of s(n) - n (With Nichole Davis and Nicole Kraght). Involve, 6(4), pp. 493--504 (2013).

I wrote this paper with two talented students in the Summer of 2013. It contains some interesting results, but the most interesting parts may be the questions that we raise. Among other things, we find several interesting families values of of s(n) - n (where s(n) represents the sum of the proper divisors of n), and other values which this difference never seems to take.

5.Reciprocal sums as a knowledge metric: theory, computation, and perfect numbers (with J. Bayless). The American Mathematical Monthly, 120(9) (November), pp. 822-831 (2013).

This is perhaps my favorite of my papers. It has a bit of history, a bit of computing, and a nice survey of a field which is little-known but really interesting. Students wanting to read one of my papers -- start here!

6.A Zeta Function for Juggling Sequences, (with C. Elsner and E. Tou). Journal of Combinatorics and Number Theory. Volume 4, Number 1, pp. 53-66.

Zeta functions have been defined in many different areas of mathematics (algebraic number theory, function fields, etc.). In this paper, we define a zeta function over the set of all juggling sequences on b balls. To do this, we first put a norm on juggling sequences. We are able to prove several things about the resulting zeta function -- among other things, we are able to find all of its zeros!

7.Generalized Collatz Functions: cycle lengths and statistics (with H. Messerman and J. Le Beau). International Journal of Undergraduate Research and Creative Activities, Volume 4, Article 2 (2012).

8.Looking for Fibonacci base-2 pseudoprimes (with D. Monfre), Missouri Journal of Mathematical Sciences, 24(2), pp. 116-123 (2011).

Pseudoprimes are composite numbers which "look prime" when you subject them to certain mathematical tests. Monfre and I were interested in those composites which look prime when subjected to the base-2 and Fibonacci primality tests. In particular, we worked to find an example of such a composite which is 2 or 3 (modulo 5). To date, no such example has been found. We extended previous searches by a factor of 200 (to 500 billion), and failed to find such a number.

9.On the Sum of the Reciprocals of Amicable Numbers (with Jonathan Bayless). Integers 11A (2011), Article 5. 17 pages.

Carl Pomerance proved in 1981 that the sum of the reciprocals of the amicable numbers converges. In this paper, Bayless and I put an upper bound on this sum (the Pomerance Constant) for the first time.

10. A Wieferich Prime Search up to 6.7 x 1015. (with Francois Dorais), Journal of Integer Sequences, 14. Article 11.9.2 (2011).

A Wieferich prime is a prime number for which 2^(p-1) = 1 (mod p^2). Two such primes (1093 and 3511) were discovered in the early twentieth century. Dorais and I developed techniques to let us quickly search for Wieferich primes, and we examined every integer up to 6.7 x 10^15, without finding any new examples. We were also able to extend our techniques to search for base-a Wieferich primes (for a=3, 5, and 7), and for Wall-Sun-Sun primes. We now hold records in the search for each of these.

11. Computing Prime Harmonic Sums, (with E. Bach and J. Sorenson) Mathematics of Computation, Volume 78, No. 268 (2009), pp. 2283--2305.

We develop and algorithm to rapdily compute the sums of the reciprocals of the primes up to x, called prime harmonic sums (in particular, the algorithm runs in about O(x^2/3) time). We then use this to answer a question of Neal Sloane, namely: what is the smallest prime p for which the prime harmonic sum up to p is greater than 4? The answer turns out to be 1801241230056600523.

Eighteenth Century Science and Mathematics

While I am quite interested in all of the History of Mathematics, the focus of my research is the Eighteenth Century, with a specific focus on the work and influence of Leonhard Euler. I've been studying Euler's work since 2002, when Lee Stemkoski and I first presented our study of the history of the 36-officer problem at the annual Euler Conference in Rumford, ME (if you're interested, read our paper!).

My primary interaction with the community of Eighteenth Century math and science scholars is through the Euler Archive, a website Stemkoski and I created in graduate school, and which I still direct. We've put scans of the vast majority (96%) of Euler's original papers and books online, and we're still working with scholars around the world in efforts to translate, summarize, and understand Euler's life and work.

My historical papers include work that is both purely historical, and mathematics inspired by historical sources.

    1. The Missing Meditatio: Leonhard Euler’s (1707–1783) contribution to articulatory phonetics (with O. Hirschey), Historiographia Linguistica, 42(1), pp. 54–73 (2015).
    2. Enlightening Symbols: A Short History of Mathematical Notation and its Hidden Powers. College Math Journal, Vol. 46, No. 1 (2015), pp. 67–72
    3. Darwin, Malthus, Süssmilch, and Euler: The Ultimate Origin of the Motivation for the Theory of Natural Selection, Journal of the History of Biology, Aug 2013, pp. 1-24 (2013).

Acknowledging that I'm biased -- this is a really fun paper. Its goal is the trace the ultimate origin of Darwin's theory of natural selection. Spoiler alert: Darwin first got the idea while reading Malthus, and from Darwin's journals we know precisely which passage he was reading. I thus traced the background of this part of Malthus's book, and found it to be based on a work of Johann Peter Süssmilch. It turns out, however, that the particular chapter of Süssmilch's work which Malthus was reading was co-authored by Euler, and based on Euler's mathematics. Thus the inspiration for the Theory of Natural Selection can be traced, ultimately, to Euler!

4. In Defense of Bertrand: the non-restrictiveness of reasoning by example, Philosophia Mathematica, 21 (3):365-370 (2013)

5. Did Euler know quadratic reciprocity? New Insights from a forgotten work. (with P. Bialek), (February 2014).

6. New Gems in Old Letters: The Pomerance-Erdös Correspondence, MAA Focus, June/July 2013.

7. An Empirical Approach to the St. Petersburg Paradox (with A. Lauren). College Mathematics Journal, Volume 4, no 2, pp. 260-264 (2012).

8.Euler Archive Settles at MAA, MAA Focus (June/July 2011), p. 12.

9. Teaching and Research with Original Sources from the Euler Archive, (with L. Stemkoski and E. Tou). Loci: Convergence (April 2011),

10. Euler and Gravity. Guest column in How Euler Did It, MAA Online, December 2009. Republished in How Euler did Even More, Mathematical Association of America, by Edward Sandifer, pp. 199-206.

11. The Euler Archive: Giving Euler to the World, (with L. Stemkoski), Euler at 300: An Appreciation. Bradley et al., eds., Mathematical Association of America, 2007, pp. 33--41. (Currently no online version available. Email me if you'd like a copy!)

12. Graeco-Latin Squares and a Mistaken Conjecture of Euler, (with L. Stemkoski), College Mathematics Journal, January 2006, pp. 2--15. Reprinted in Dunham, William. (Ed.) The Genius of Euler: Reflections on his Life and Work.

Euler believed that a Greco-Latin square of size 6 could not be constructed. More generally, he thought this held for any size n = 4k+2. He was correct on the first point, but very wrong on the second. Interestingly, it took

13. I also write a quarterly column, "The Omnipresent Savant", for Opusculum, the newsletter of the Euler Society. They can all be accessed at the society's webpage using this link.

    1. Euler, Darwin, and Population Statistics, Opusculum, Volume 1, Issue 2 (2009), pp. 4--6.
    2. On Citing Euler's works, Opusculum, Volume 1, Issue 3 (2009), pp. 8--10.
    3. Opera Omnia Anglicae Reddita, Opusculum, Volume 2, Issue 1 (2010), pp. 1-4.
    4. Euler as Master Teacher in the Letters to a German Princess, Opusculum, Volume 2, Issue 2 (2010), pp. 17-21.
    5. Euler's Letters to a German Princess: Betrayal and Translation, Opusculum, Volume 3, Issue 1 (2011), pp. 23-26.
    6. Seeking the original text of Euler's Letters to a German Princess, Opusculum, Volume 3, Issue 2 (2011), pp. 30-32.
    7. Español, Português, and Català: Eulerian scholarship in modern Romance languages. Opusculum, Volume 3, Issue 3 (2012).

Additionally, I've been working with students and other faculty to study other, non-mathematical aspects of Euler's work. See Current Projects for more information about this.

Applied Statistics and Interdisciplinary Work

One of my favorite activities to find ways to apply mathematics to new fields, and to look for connections between mathematics and other fields. More more details, see my Current Projects page.

  1. Optimum Injury and Illness Prevention Cost for U.S. Construction Projects (with S. Rajendran and M. Bliss), to appear in Practice Periodical on Structural Design and Construction.
  2. Writing About Shakespeare: 1960-2010 (with L. Estill), Journal of Open Humanities Data, 2, p.e3 (2016). DOI: http://doi.org/10.5334/johd.4
  3. Spare your arithmetic, never count the turns: A Statistical Analysis of Writing about Shakespeare, 1960-2010 (with K. Bridal and L. Estill), Shakespeare Quarterly, 66(1), Spring 2015, pp. 1–28 (2015).

Gastroenterology

    1. A Prospective Study on Endoscopy for Luminal Abnormalities on Imaging and Its Impact on
    2. Clinical Management (with V. Kumaravel, M. Mahmoud, L. Hernandez, and N. Guda), abstract, Gastrointestinal Endoscopy 81 (5), pp. AB237–AB238 (2015).
    3. Temporal Trends in the Detection of Right-Sided Colon Polyps: Summary Data of Large GI Practices (with N. Guda, N. Gupta, J. Allen, T. Deas, L. Huang, S. Ketover, K. Etzkorn, K. Gutta, S. Morris, P. Sharma, and L. Hernandez), abstract. Gastrointestinal Endoscopy 01/2014; 79(5): AB117.
    4. A pilot study of endoluminal US for stool liquefaction (with L. Hernandez, G. Triadafilopoulos, J. Kost, R. Ganz, S. Fleshman, M. Ton, and G. Lewis. Gastrointestinal Endoscopy, 3/14 79(3), pp. 508–513, 2013.
    5. Colonoscopy Quality Indicators From a Nationwide Consortium of GI Practices, with (L. Hernandez, J. Allen, J. Geenen, M. Schmalz, M. Catalano). Gastrointestinal Endoscopy, Volume 73, Issue 4, Supplement, Page AB396, (2011).

This is part of an ongoing study to compare the quality of GI Practices, adjusting for various differences in the demographics of patient populations.

6. Malpractice claims for endoscopy (with L. Hernandez and S. Regenbogen), World Journal of Gastrointestinal Endoscopy 5(4): 169-73, 2013.

7.A Longitudinal Assessment of Colonoscopy Quality Indicators: a Report From the Gastroenterology Practice Management Group (Gpmg), (with L. Hernandez, J. Allen, T. Deas, N. Guda, M. Schmalz, and M. Catalano), Gastrointestinal Endoscopy 77 (5), p. AB127 (2013).

8. Salvaging Poorly Prepared Colonoscopies Using Endoluminal Ultrasound: a Pilot Study (with L. Hernandez, G. Triada.lopoulos, J. Kosh, R. Ganz, M. Ton, and G. Lewis.) Gastrointestinal Endoscopy 77 (5), pp. AB429-AB430.

9. Quality-adjusted life expectancy benefits of laparoscopic bariatric surgery: A United States Perspective, (with L. Hernandez), International Journal of Technology Assessment in Health Care 26:3, pp. 280--287 (2010). (no online version available, email me if you'd like to read a copy!)

Hernandez and I compare the two most common types of baryatric surgery, commonly called banding and bypass, to determine their long-term effects in terms of quality-adjusted life expecatancy. We use a Markov model to follow simulated cohorts through life, tracking age, body mass index, and health.

Other

Expository pieces

  1. Media Exposure on Student Work: Spotlight on Undergraduate Research, to appear in PRIMUS (2016).
  2. New Gems in Old Letters: The Pomerance-Erdős Correspondence, MAA Focus, June/July 2013, pp. ??
  3. Where Math Meets Art: new frontiers in the mathematics of juggling, eJuggle: The offi.cial publication of the International Jugglers' Association, September 2012.
  4. Euler Archive Settles at MAA, MAA Focus (June/July 2011), p. 12.

Math Education

  1. Measuring Habits of Mind: Toward a Prompt-less Instrument for Assessing Quantitative Literacy (with S. Boersma), Numeracy, Volume 6, Issue 1, Article 6.
  2. Teaching Quantitative Reasoning as an Honors Course (with S. Boersma), Numeracy, Volume 6, Issue 1, Article 6.
  3. Using Technology to Reform a Calculus Course, (with J. Korey and D. Lahr), Proceedings of the 11th Annual Conference of the South African Association for Research in Mathematics, Science, and Technology Education, 2003, pp. 716-721.