Enrique Treviño, Ph. D.

Phone: 
Ext.#6187 
Office: 
Brown Hall 123 
Office Hours: 
Spring 2023: MWF from 10am to 121:45am, and by appointment. 
Email: 
trevino@lakeforest.edu

About Me
I am an associate professor of Mathematics at Lake Forest College. I started working as an Assistant Professor in 2013 and was promoted in 2018. I got a baccalaureate of Science degree in Mathematics from the University of Texas at
El Paso (UTEP) in 2006. I completed my PhD in Mathematics at Dartmouth College in June 2011, working with Carl Pomerance. My main research interest is Number Theory.
My dissertation is titled Numerically Explicit Estimates for Character Sums; it involves a
blend of analytic number theory and some serious computing.
I continue to be interested in these kinds of problems, such as estimating the least quadratic nonresidue,
the least inert prime in a real quadratic field, and other concrete inequalities. Besides my interest in number theory, I also write recreational mathematics papers.
I am very fond of mathematical competitions such as the Mathematical Olympiad. I have been involved either as student or as a teacher in the Olympiad since the year
2000. Since 2015, I have been a member of the Mexican Mathematical Olympiad Committee. My love for problem solving extends to other competitions such as the Putnam,
and to problem solving publications (such as The Riddler in FiveThirtyEight and the problem section in mathematics journals.)
My current editorial duties are: I am an associate editor of the College Mathematics Journal and of the Boletín de la Sociedad Matemática Mexicana. I am also coeditor in chief of the United States Mathematical Olympiad.
Curriculum Vitae and other materials
Curriculum Vitae
Research Statement
Teaching Statement
Current Classes
Math 109: Calculus 1b Lake Forest College Spring 2023.
Math 111: Calculus II Lake Forest College Spring 2023.
Math 499: Senior Seminar: Great Theorems of Mathematics Lake Forest College Spring 2023.
Classes I've Taught
Math 108: Calculus 1a Lake Forest College Fall 2022.
Math 110: Calculus I Lake Forest College Fall 2022.
Math 230: Abstract and Discrete Mathematics Lake Forest College Fall 2022.
Math 340: Geometry Lake Forest College Fall 2022.
Math 110: Calculus I Lake Forest College Summer 2022.
Math 108: Calculus 1a Lake Forest College Fall 2021.
Math 230: Abstract and Discrete Mathematics Lake Forest College Fall 2021.
Math 323: Cryptography Lake Forest College Fall 2021.
Math 110: Calculus I Lake Forest College JTerm 2021.
Math 110: Calculus I Lake Forest College Summer 2020.
Math 115: Honors Calculus I Lake Forest College Summer 2020.
Math 109: Calculus 1b Lake Forest College Spring 2020.
Math 231: Linear Algebra Lake Forest College Spring 2020.
Math 330: Abstract Algebra Lake Forest College Spring 2020.
Math 108: Calculus 1a Lake Forest College Fall 2019.
Math 230: Abstract and Discrete Mathematics Lake Forest College Fall 2019.
Math 323: Cryptography Lake Forest College Fall 2019.
Math 110: Calculus I Lake Forest College Summer 2019.
Math 115: Honors Calculus I Lake Forest College Summer 2019.
Math 109: Calculus 1b Lake Forest College Spring 2019.
Math 331: Modern Algebra II Lake Forest College Spring 2019.
Math 375: Combinatorics Lake Forest College Spring 2019.
Math 108: Calculus 1a Lake Forest College Fall 2018.
Math 110: Calculus I Lake Forest College Fall 2018.
Math 330: Modern Algebra I Lake Forest College Fall 2018.
Math 499: Senior Seminar: Great Theorems of Mathematics Lake Forest College Fall 2018.
Math 110: Calculus I Lake Forest College Summer 2018.
CS 112: Computer Science I Lake Forest College Spring 2018.
Math 230: Abstract and Discrete Mathematics Lake Forest College Spring 2018.
Math 329: Number Theory Lake Forest College Spring 2018.
CS 112: Computer Science I Lake Forest College Fall 2017.
Math 311: Introduction to Real Analysiss Lake Forest College Fall 2017.
Math 499: Senior Seminar: Great Theorems of Mathematics Lake Forest College Fall 2017.
Math 210: Multivariable Calculus Lake Forest College Fall 2016.
Math 340: Geometry Lake Forest College Fall 2016.
Math 499: Senior Seminar: Great Theorems of Mathematics Lake Forest College Fall 2016.
Math 110: Calculus I Lake Forest College Summer 2016.
Math 150: Introduction to Probability and Statistics Lake Forest College Spring 2016.
Math 330: Modern Algebra I Lake Forest College Spring 2016.
Math 411: Real Analysis II Lake Forest College Spring 2016.
FIYS 169: Recreational Mathematics Lake Forest College Fall 2015.
Math 210: Multivariable Calculus Lake Forest College Fall 2015.
Math 230: Abstract and Discrete Mathematics Lake Forest College Fall 2015.
Math 110: Calculus I Lake Forest College Summer 2015.
Math 160: Math Methods with Applications Lake Forest College Summer 2015.
Math 150: Introduction to Probability and Statistics Lake Forest College Spring 2015.
Math 214: Differential Equations Lake Forest College Spring 2015.
Math 230: Abstract and Discrete Mathematics Lake Forest College Spring 2015.
FIYS 169: Recreational Mathematics Lake Forest College Fall 2014.
Math 330: Modern Algebra I Lake Forest College Fall 2014.
Math 160: Math Methods with Applications Lake Forest College Summer 2014.
Math 150: Introduction to Probability and Statistics Lake Forest College Spring 2014.
Math 230: Abstract and Discrete Mathematics Lake Forest College Spring 2014.
Math 160: Math Methods with Applications Lake Forest College Fall 2013.
Math 230: Abstract and Discrete Mathematics Lake Forest College Fall 2013.
Math 25: Further Topics Single Variable Calculus Swarthmore College Spring 2013.
Math 53: Analytic Number Theory Swarthmore College Spring 2013.
Math 28: Linear Algebra Honors Course Swarthmore College Fall 2012.
Math 58: Number Theory Swarthmore College Fall 2012.
Math 25: Further Topics Single Variable Calculus Swarthmore College Spring 2012.
Math 77: Advanced Topics in Algebra: Algebraic Number Theory Swarthmore College Spring 2012.
Math 27: Linear Algebra Swarthmore College Fall 2011.
Math 20: Discrete Probability, Dartmouth College Summer 2009.
Math 2: Calculus with Algebra and Trigonometry, Dartmouth College Winter 2009.
Papers
Clicking on the links below takes you to a preprint of the paper. Undergraduate (at the time) authors are shown with an asterisk.
 Introducción a la Teoría de Números Probabílistica.
Submitted.
 Partitioning powers into sets of equal sum
with Paul Pollack. Submitted.
 Many proofs that 1+2+... = n(n+1)/2
with Tom Edgar. Published in Ganit Bikash
(translates to Mathematics Development) an Assamese journal, Vol. 75 (2022), pp. 3861. The version in English is here.
 Generalizing Parking Functions with Randomness
with M. Tian*. Electronic Journal of Combinatorics Vol. 29 (2022), no. 3, P3. 13, 13 pages (electronic).
 On the sum of kth powers in terms of earlier sums
with S. J. Miller. The College Mathematics Journal, Vol. 53 (2022), no. 3, pp. 220225.
 The least quadratic nonresidue
with K. McGown. Mexican Mathematicians in the World: Trends and recent contributions published by Contemporary Mathematics of the AMS (2021), pp. 205232.
 On a sequence related to the factoradic representation of an integer
with M. Sánchez Garza*. Journal of Integer Sequences, Vol. 24 (2021), Article 21.8.5, 13 pages (electronic).
 On sets whose subsets have integer mean.
Integers 21 (2021), article A79, 11 pages (electronic).
 On Egyptian fractions of length 3
with C. Banderier, C. A. Gómez Ruiz, F. Luca, and F. Pappalardi. Revista de la Unión Matemática Argentina, Vol. 62 (2021), no. 1, pp. 257274.
 An unusual recursive formula to answer a question
regarding fixed points in permutations
with M. Tian*. The College Mathematics Journal vol. 52 (2021), no. 3, pp. 219220.
 A writingintensive FYS course on Recreational Mathematics.
Published as Chapter 30 in book "Mathematical themes in a firstyear seminar" published by MAA Press (2021), pp. 301306.
 Sums of proper powers
with Paul Pollack. The American Mathematical Monthly vol. 128 (2021), no.1, p. 40.
 On sums of consecutive triangular numbers
with P. Pollack and D. Subramaniam*. Integers 20A (2020), article A15, 10 pages (electronic).
 Expected number of dice rolls for the sum to reach n.
American Mathematical Monthly vol. 127 (2020), no. 3, p. 257.
 On generalizing happy numbers to fractional base number systems
with M. Zhylinski*. Involve vol. 12 (2019), no. 7, pp. 11431151.
 Probabilistic Proof that 1+2+...+n = n(n+1)/2.
American Mathematical Monthly vol. 126 (2019), no. 9, p. 840
 A birthday in St. Petersburg.
The College Mathematics Journal vol. 50 (2019), no. 1, pp. 3640.
 An InclusionExclusion Proof of Wilson's Theorem.
The College Mathematics Journal vol. 49 (2018), no. 5, pp. 367368.
 Walking on rational numbers and a selfreferential formula
with M. Fortman*, K. Kupiec*, and M. Rawlings*. Elemente der Mathematik 73 (2018), no. 4, 161169.
 A short proof of a sum of powers formula
American Mathematical Monthly 125 (2018), no. 7, 659659.
 Finding the four squares in Lagrange's Theorem
with Paul Pollack. Integers 18A (2018), article A15, 16 pages (electronic).
 Counting Perfect Polynomials
with U.C. Cengiz* and P. Pollack. Finite Fields and Their Applications 47C (2017) pp. 242255 DOI information: 10.1016/j.ffa.2017.05.006.
 Resolving Grosswald's conjecture on GRH
with Kevin McGown and Timothy Trudgian. Functiones et Approximatio, Commentarii Mathematici 55.2 (2016), pp. 215225.
 The Burgess inequality and the least kth power nonresidue
International Journal of Number Theory Vol. 11, No. 5 (2015), pp. 126.
 The smoothed PólyaVinogradov inequality
with K. Adamczewski*, Integers 15 (2015), article A20, 11 pages (electronic).
 The least kth power non residue
Journal of Number Theory 149 (2015), pp. 201224. DOI information: 10.1016/j.jnt.2014.10.019
 The primes that Euclid forgot
with P. Pollack, American Mathematical Monthly 121 (2014), no. 5, 433437.
 Sets of monotonicity for Euler's totient function
with P. Pollack and C. Pomerance, The Ramanujan Journal 30 (2013), no. 3, 379398.
 On the maximum number of consecutive integers on which a character is constant
Moscow Journal of Combinatorics and Number Theory 2012, vol.2, iss. 1, pp. 5672. Corrigendum appeared in Moscow Journal of Combinatorics and Number Theory 2017, vol.7, iss. 3.
 The least inert prime in a real quadratic field
Mathematics of Computation, vol. 81, no. 279, July 2012, pp. 17771797.
 Multidimensional Frobenius problem
with J. Amos*, I. Pascu*, V. Ponomarenko and Y. Zhang*, Involve, a Journal of Mathematics 42 (2011), 187197.
 On the counting function for the generalized Niven numbers
with R. Daileda, J. Jou*, R. LemkeOliver* and E. Rossolimo*, J. Théor. Nombres Bordeaux 21 (2009), no. 3, 503515.
Books
Clicking on the link takes you to a website where you can purchase the book.
Cuban Mathematical Olympiads written by Roberto Bosch and translated by yours truly. Published by XYZ Press in 2017. ISBN13: 9780996874540.
Problems
I have invented problems that have appeared in competitions or in math journals. I compiled them below:
Invented Problems
Student Posters, Papers and Presentations
Kevin Kupiec and Marina Rawlings worked with me on a 3week research project as Richter scholars in May 2014. The project was titled
What's so rational about the alphabet?. They presented the following poster at the Richter Symmposium in
June 2014. The following are the slides they made for a presentation at the Lake Forest student Symposium in
April 2015. This work together with Margaret Fortman's (mentioned below) is the content of the paper Walking on rational numbers and a selfreferential formula.
Ugur Caner Cengiz worked with me on a 10week research project as a Richter scholar in the Summer of 2014. He wrote the following paper:
Finding perfect polynomials mod 2. The following are the
slides for the presentation he gave at the Lake Forest Student Symposium. This work, together with some results by Paul Pollack formed the content of the paper Counting perfect polynomials.
Margaret Fortman worked with me on a 4week research project as a Richter scholar in the Summer of 2015. She made the following poster on her work on the Tupper selfreferential formula:
Poster. Margaret also wrote a paper about her work. You can find it here. This work together with the work done by Kevin Kupiec and Marina Rawlings is the content of the paper Walking on rational numbers and a selfreferential formula.
Robert Mecham worked with me on a 10week research project as a Richter scholar in the summer of 2015. Robert worked on the problem of placing firstyear students to a firstyear studies class that they rank highly
while balancing constraints. His presentation "Firstworld solutions to Firstyear problems" was given at the Richter Symposium in July 2015.
Noel Orwothwun worked with me on a 4week research project as a Richter scholar in the summer of 2016. Noel worked on Beatty sequences and prime
number races on them. His talk Beatty Sequences and the Prime Race was given at the Richter Symposium in June 2016.
Dipika Subramaniam worked with me on a 4week research project as a Richter scholar in the summer of 2018. Dipika worked on finding sums of triangular numbers that add to a triangular number.
Her poster Playing with Triangular Numbers was presented at the Richter Symposium in June 2018.
Christian Reyes worked with me on a 4week research project as a Richter scholar in the summer of 2019. Christian worked on writing an algorithm to sort students into Richter projects
based on their preferences. He presented the following at the Richter Symposium on June 2019: Solving Stable Matching Problems With the
GaleShapley Algorithm.
Arthur J. Brown worked with me on a 4week research project as a Richter scholar in the summer of 2019. AJ worked on establishing whether a model for consociational democracy showed estability. The project
built on work Finnian Bunta did for his undergraduate thesis in 2018. AJ's poster The Mathematics of Consociational Democracy was presented at the Richter Symposium in June 2019.
Student Undergrad Thesis
The probabilistic method by Melanie Tian, Fall 2021.
The Fundamental Theorem of Algebra by William Braubach, Fall 2016.
Transcendental Numbers by Jacob Juillerat, Spring 2016.
Saltos pequeños entre primos by Juan Ramón Camacho Cordero, Universidad de Guanajuato, 2012.
Recreational Mathematics
Generalizando un problema de la OMM 2017 This paper generalizes a problem from the 31st Mexican Math Olympiad (2017). I eventually expanded these results into the paper "On sets whose subsets have integer mean".
Many Proofs of the formula for sum of consecutives I compiled many proofs that 1+2+...+n = n(n+1)/2 and then Tom Edgar added some more.
Underhand Free Throws This is my solution to a riddle in the website FiveThirtyEight.
The riddle has to do with how much better a player would be by shooting underhand free throws instead of the normal kind. One nice aspect about this pdf is how I use Riemann sums to figure out
the joint probability distribution.
A direct combinatorial approach to the sum of the kth
powers of the first n integers This is a longer piece than the one I submitted for publication. It includes examples for the sum 1+2+...+n and 1^2+...+n^2.
A result about random variables inspired by the book "The Simpsons and their mathematical secrets". Note: Pages 3 and 4 are a proof of the same result as on the first 2 pages but without using integrals.
I eventually edited this down into the paper "Expected number of dice rolls for the sum to reach n" that appeared in the Monthly in 2020.
A bijective proof of a simple mutliplicative number theory result. Note: The theorem is easy to prove without the bijection but maybe not as insightful.
A pair of surprising identities. Note: I wrote these proofs after Walter Stromquist mentioned these identities to me. My proofs, as you can see, are not that well written. I apologize for that.
Facebook Friends. Note: I recommend reading Steven Strogatz NYTimes article on this phenomenon.
On the number of coin flips needed to get the same number of heads as tails. Note: My colleague Sugata Banerji mentioned this as a programming exercise and I liked the problem (specially with the appearance of Catalan numbers).
I think it also has some similarities with the St. Petersburg paradox. I submitted a different version titled "A birthday in St. Petersburg" which appeared in the College Mathematics Journal in 2019.
A proof of a stronger Law of Sines using the Law of Cosines. An unnatural use of the law of cosines to prove the law of sines.
AwesomeMath Program
I work at the Awesome Math summer program in the summers teaching fun mathematics to middle school and high school students from around the country
(and the world). It is a great 3 week program (offered three times per summer) with classes of varying difficulty. The program was
funded by Titu Andreescu, who has authored many problemsolving books and was the leader of the USA IMO (International Math Olympiad) team for many years.
Picture with my Geometry 1 students at the University of Puget Sound in 2016
Talks
The least quadratic nonresidue and related problems given (virtually) at the LFunctions in Analytic Number Theory Collaborative Research Group Seminar (organized as part of the Pacific Institute for Mathematical Sciences in Canada) on January 18, 2023.
Partitioning powers into sets of equal sum given at a BSM Special Session on Mathematical Research in Budapest for Students and Faculty at the Joint Math
Meetings in Boston on January 5, 2023.
El mínimo norediduo cuadrático y otros problemas relacionados given (virtually) at the Semana Matemática at Universidad Autónoma de Santo Domingo on October 4, 2022.
Partitioning powers into sets of equal sum given at an AMS Special Session on Algebraic Structures in Topology, Logic, and Arithmetic at the AMS Fall Central Sectional Meeting, at the University of Texas at El Paso on September 17, 2022.
Partitioning powers into sets of equal sum given at the Budapest Semesters in Mathematics program in Budapest on July 5, 2022.
Paradojas Matemáticas en probabilidad given at EMALCA (Escuela de Matemática de América Latina y el Caribe) at the Universidad Autónoma de Santo Domingo on June 3, 2022.
Tres proyectos de investigación con estudiantes de licenciatura given virtually at Seminario Coloquio GRACIARed Matemática (República Dominicana) on April 1, 2022.
Polymath REU: A Program to Encourage
Undergraduate Math Research Across the Globe given at the Lake Forest College Faculty Discussion Group on November 15, 2021.
A trio of research projects with undergraduates given virtually at the Colloquium of on Smith College October 21, 2021.
A trio of research projects with undergraduates given virtually at the Colloquium of Hofstra University on September 29, 2021.
Avances recientes sobre la distribución de los números primos given virtually at the Colloquium of the Universidad Estatal a Distancia (Costa Rica) on March 24, 2021.
On sets whose subsets have integer mean given virtually at the Illinois MAA sectional meeting on March 12, 2021.
Partitioning powers into sets of equal sum given virtually at the Joint Math Meetings at the AMS Special Session on A Showcase of Number Theory at Undergraduate Institutions given on January 6, 2021.
Cumpleaños en San Petersburgo given virtually at the Seminario Unplugged de Geometría at the Universidad de Colima on November 27, 2020.
Paradojas Matemáticas given virtually for the Fundapromat on September 21, 2020.
Egyptian equations and modern mathematics given at the Lake Forest College Faculty Discussion on January 29, 2020.
On Egyptian Fractions of length 3 given at the Number Theory Down Under 7 conference held at the University of New South Wales in Sydney, Australia on October 3, 2019.
Playing with Triangular Numbers given at the University of Texas at El Paso colloquium on February 15, 2019.
Playing with Triangular Numbers given at the West Coast Number Theory Conference on December 16, 2018.
Playing with Triangular Numbers given at the California State University at Chico on December 13, 2018.
El mínimo norediduo cuadrático y otros problemas relacionados given at the Fourth Reunion of Mexican Mathematicians around the World at the Casa Matemática Oaxaca on June 11, 2018. A video of the talk can be seen here.
Counting Perfect Polynomials given at the IndianaIllinoisMichigan MAA trisectional conference on March 23, 2018.
Finding the four squares in Lagrange's theorem given at the Joint Math Meetings at the AMS Special Session on A Showcase of Number Theory at Liberal Arts Colleges given on January 11, 2018.
Counting Perfect Polynomials given at the West Coast Number Theory Conference on December 18, 2017. Winner of the Lehmer Prize
Gerrymandering and Math given at the Faculty Discussion Group at Lake Forest College on October 26, 2017.
Walking on numbers and a selfreferential formula given at the Awesome Math Summer Camp at Cornell University on August 3, 2017.
The least quadratic nonresidue and related problems given at the Séminaire Dynamique, Arithmétique, Combinatoire of the I2M Math Laboratory of AixMarseille Université on March 28, 2017.
Summer Research Projects for First Year Students given at the AMS Special Session on Open & Accessible Problems for Undergraduate Research at the Joint Math Meetings on January 7, 2017.
Resolving Grosswald's conjecture assuming GRH given at the West Coast Number Theory Conference on December 17, 2016.
Summer Research Projects for First Year Students given at the California State University at Chico on October 17, 2016.
Resolving Grosswald's conjecture on GRH given at the Carl Pomerance's 70th birthday Conference on June 11, 2015.
The least quadratic nonresidue and related problems given at the California State University at Chico on April 17, 2015.
Prime gaps: a breakthrough in number theory given at the Faculty Discussion Group at Lake Forest College on September 16, 2014.
El mínimo norediduo cuadrático y otros problemas relacionados given at the Instituto de Ciencias Matemáticas in the Universidad Autónoma de Madrid on July 21, 2014.
The Burgess inequality and the least kth power nonresidue given at the Emerging New Faces in Analytic Number Theory (ENFANT) workshop at the Hausdorff Institute on July 12, 2014.
The least quadratic nonresidue modulo a prime and related problems given at the Midwest Number Theory Conference for Graduate Students and Recent PhDs on June 3, 2014.
The primes that Euclid forgot given at the Oberlin Number Theory Seminar on March 10, 2014.
Prime Gaps: A breakthrough in number theory given at the Seminario Interuniversitario de Investigación en Ciencias de Matemáticas in Ponce, Puerto Rico on March 1, 2014.
The primes that Euclid forgot given at the AMS Special Session on Analytic Number Theory at the Joint Meetings on January 16, 2014..
Bounds on graphs with high girth and high
chromatic number given at the INTEGERS Erdos Centennial Conference on Combinatorial Number Theory on October 26, 2013.
The primes that Euclid forgot given at the Underrepresented Students in Algebra and Topology Symposium on April 20, 2013.
On squares and nonsquares modulo a prime given at Lake Forest College on February 12, 2013.
The Burgess inequality and the least kth power nonresidue. given at the Special Session on Arithmetic Statistics at the Joint Meetings of the AMS/MAA on January 10, 2013.
The numerically explicit Burgess inequality and an application to quadratic nonresidues. given at the AMS Sectional Meeting in Akron, OH on October 21, 2012.
El mínimo norediduo cuadrático y otros problemas relacionados given at a colloquium lecture at the Universidad de Colima on May 8, 2012
On the maximum number of consecutive
integers on which a character is constant given at the Joint Math Meetings on January 5, 2012.
The smoothed PólyaVinogradov inequality given at the Integers Conference, October 28, 2011.
The least quadratic nonresidue and related problems given at a colloquium lecture at Swarthmore College on March 1st, 2011, at University of North Texas on October 2011 and at the automorphic forms seminar at Purdue University in February 2012.
The least inert prime in a real quadratic field given at the Palmetto Number Theory Series XIV, December 4, 2010.
Burgess bounds for the Burgess inequality for character sums given at the Integers conference, October 16, 2009.
On the counting function of the generalized Niven numbers given at the Québec/Maine Number Theory Conference, October 2008.
The higherdimensional Frobenius problem given with Yan Zhang at Young Mathematician's Conference (YMC), August 2005.
Mathematical Olympiads given at a colloquim lecture at Trinity University on July 2005.
Links
Olimpiada Mexicana de Matemáticas Chihuahua
Last modified on January 18, 2023 by Enrique.