Associate Professor of Mathematics
Chair of the Mathematics Department
Cedar Crest College
Office: Curtis 221 Email:Joshua.Harrington@cedarcrest.edu Calendar:Outlook
On the iteration of a function related to Euler's phi-function with Lenny Jones, Integers 10, 2010.
On the Factorization of the Trinomials x^n+cx^(n-1)+d, Int. J. Number Theory 8, 2012, 1513-1518.
Arithmetic progressions in the polygonal numbers with Scott Dunn and Kenneth Brown, Integers 12, 2012.
A polynomial investigation inspired by work of Schinzel and Sierpinski with Michael Filaseta, Acta Arith. 155, 2012, 149-161.
Nonlinear Sierpinski and Riesel Numbers with Carrie Finch and Lenny Jones, J. Number Theory 133, 2013, 534-544.
A class of irreducible polynomials with Lenny Jones , Colloq. Math. 132, 2013, 113-119.
The reducibility of constant-perturbed products of cyclotomic polynomials with Lenny Jones and Daniel White, Int. J. Number Theory 10, 2013, 13-29.
The Factorization of f(x)x^n+g(x) when deg f\leq 2 with Andrew Vincent and Daniel White, J. Theor. Nombres Bordeaux 25, no. 3, 2013, 565-578.
Representing integers as the sum of two squares in the ring Zn with Lenny Jones and Alicia Lamarche, J. Integer Seq. 17, 2014, no. 7, Article 14.7.4.
Coloring of Pythagorean triples within colorings of the positive integers with Joshua Cooper, Michael Filaseta, and Daniel White, J. Comb. Number Theory, 2014, Volume 6, Issue 1.
Characterizing finite groups using the sum of the orders of the elements with Lenny Jones and Alicia Lamarche, Int. J. Comb., 2014, Art. ID 826141, 11pp.
Extending a theorem of Pillai to quadratic sequences with Lenny Jones, INTEGERS 15A.
Differences Between Elements of the Same Order in a Finite Field, with Lenny Jones, J. Number Theory 180, 2017, 443-459
An equation involving arithmetic functions and Riesel numbers, with Lenny Jones, Integers 18, 2018.
A new condition equivalent to the Ankeny-Artin-Chowla conjecture, with Lenny Jones, J. Number Theory 192, 2018, 240-250.
A modification of a problem of Diophantus, with Lenny Jones, Math. Slovaca 68, no. 6, 2018, 1343-1352.
Odd coverings of subsets of the integers, with James Hammer and Kristina Marotta, J. Comb. Number Theory, 2018, Volume 10, Issue 2.
An investigation on partitions with equal products, with Byungchul Cha, Adam Claman, Ziyu Liu, Barbara Maldonado, Alexander Miller, Ann Palma, Tony W. H. Wong, and Hongkwon Yi, Int. J. Number Theory 15 (2019), no. 8, 1731-1744.
On super totient numbers and super totient labelings of graphs, with Tony W.H. Wong, Discrete Math. 343 (2020), no. 2, 111670, 11 pp.
Graph polynomials for a class of DI-pathological graphs, with James Hammer, AKCE Int. J. Graphs. Comb. 17 (2020), no. 1, 206-212.
Monogenic binomial compositions, with Lenny Jones, Taiwanese J. Math. 24 (2020), no 5., 1073-1090.
On the domination number of permutation graphs and an application to strong fixed points, with Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Daniel McGinnis, Riley Waechter, and Tony W. H. Wong, Discrete Appl. Math. 288 (2021), 20-34.
Two dependent probabilistic chip-collecting games with Kedar Karhadkar, Madeline Kohutka, Tessa Stevens, and Tony W. H. Wong, Discrete Appl. Math. 288 (2021), 74-86.
On binomial coefficients associated with Sierpinski and Riesel numbers, with Ashley Armbruster, Grace Barger, Sofya Bykova, Tyler Dvorachek, Emily Eckard, Yewen Sun, and Tony W. H. Wong, Integers 21, 2021.
Submitted Research
Monogenic cyclotomic composotions, with Lenny Jones.
Sums of distinct cubic polynomial residues, with Carrie Finch-Smith and Tony W. H. Wong.
Sum index and difference index of graphs, with Eugene Henninger-Voss, Kedar Karhadkar, Emily Robinson, and Tony W. H. Wong.
Consecutive Sierpinski numbers, with R. Groth and Carrie Finch-Smith.
Some new polynomial discriminants, with Lenny Jones.
On k-fold super totient numbers, with Melea Roman and Tony W. H. Wong.
Exceptional Totient Numbers, with Breille Duncan and Andrew Vincent.
Covering systems with odd moduli, with Yewen Sun and Tony W. H. Wong.
On properties of fibotomic polynomials, with Cameron Byer, Tyler Dvorachek, Emily Eckard, Linsey Wise, and Tony W. H. Wong.
Contributions to the Online Encyclopedia of Integer Sequences
Number of units u in Z/nZ such that Phi(3,u) is a unit, where Phi is the cyclotomic polynomial, A289460, with Eric Jovinelly, Jordan Lenchitz, Michael Mueller, Tristan Phillips, and Madison Wellen.
Number of units u in Z/(2n-1)Z such that Phi(4,u) is a unit, where Phi is the cyclotomic polynomial, A289835, with Eric Jovinelly, Jordan Lenchitz, Michael Mueller, Tristan Phillips, and Madison Wellen.
Number of units u in Z/nZ such that Phi(5,u) is a unit, where Phi is the cyclotomic polynomial, A290309, with Eric Jovinelly, Jordan Lenchitz, Michael Mueller, Tristan Phillips, and Madison Wellen.
Sum modulo n of all units u in Z/nZ such that Phi(3,u) is a unit, where Phi is the cyclotomic polynomial, A290321, with Eric Jovinelly, Jordan Lenchitz, Michael Mueller, Tristan Phillips, and Madison Wellen.
Sum modulo n of all units u in Z/nZ such that Phi(5,u) is a unit, where Phi is the cyclotomic polynomial, A290322, with Eric Jovinelly, Jordan Lenchitz, Michael Mueller, Tristan Phillips, and Madison Wellen.
A triple of positive integers (n,p,k) is admissible if there exist at least two different multisets of k positive integers, {x_1,x_2,...,x_k} and {y_1,y_2,...,y_k}, such that x_1+x_2+...+x_k=y_1+y_2+...+y_k = n and x_1x_2...x_k = y_1y_2...y_k = p. For each n, let A(n)={k:(n,p,k) is admissible for some p}, and let a(n) = |A(n)|, A316945, with
Byungchul Cha, Adam Claman, Ziyu Liu, Barbara Maldonado, Alexander M. Miller, Ann Palma, Wing Hong Tony Wong, and Hongkwon V. Yi.
A triple of positive integers (n,p,k) is admissible if there exist at least two different multisets of k positive integers, {x_1,x_2,...,x_k} and {y_1,y_2,...,y_k}, such that x_1+x_2+...+x_k = y_1+y_2+...+y_k = n and x_1x_2...x_k = y_1y_2...y_k = p. For each n, let A(n) = {p:(n,p,k) is admissible for some k}, and let a(n) = |A(n)|, A316946, with
Theresa Baren, James Hammer, Ziyu Liu, Sean E. Rainville, Melea Roman, and Hongkwon V. Yi.
For n>=3, smallest prime number N such that for every prime p>=N, every element in Z_p can be expressed as a sum of two n-gonal numbers mod p, without allowing zero as a summand, A317244, with
Byungchul Cha, Adam Claman, Ziyu Liu, Barbara Maldonado, Alexander M. Miller, Ann Palma, Wing Hong Tony Wong, and Hongkwon V. Yi.
a(n) is the smallest integer such that for all s >= a(n), there are at least n-1 different partitions of s into n parts, namely {x_{11},x_{12},...,x_{1n}}, {x_{21},x_{22},...,x_{2n}},..., and {x_{n-1,1},x_{n-1,2},...,x_{n-1,n}}, such that the products of every set are equal, A317254, with
Byungchul Cha, Adam Claman, Ziyu Liu, Barbara Maldonado, Alexander M. Miller, Ann Palma, Wing Hong Tony Wong, and Hongkwon V. Yi.
Numbers that are not divisible by the product of their nonzero digits, A346657, with Michael Gohn, Sophia Lebiere, Hani Samamah, Kyla Shappell, and Tony W. H. Wong.
Presentations
Bloomsburg University
Investigating the Factorization of Monic Trinomials, Seminar Series, June 2012.
Canadian Number Theory Association (CNTA)
On the Factorization of the Trinomials x^n+cx^(n-1)+d, CNTA XII, June 2012.
Clemson University
Invited speaker, Covering Systems and Some Applications, Number Theory Seminar, Fall 2012.
INTEGERS 2013: The Erdos Centennial Conference
Invited speaker, A Polynomial Investigation Inspired By Work of Schinzel and
Sierpinski, October 2013.
Joint Mathematics Meetings
Odd Coverings of Subsets of the Integers, January 2019.
Sums of Polynomial Residues Mod n, January 2017.
Special Numbers in Z_n, January 2016.
The Reducibility of Constant-Perturbed Products of Cyclotomic Polynomials,
January 2014.
Invited speaker, A Polynomial Investigation Inspired by Work of Schinzel and
Sierpinski, January 2013.
On the Iterations of a Function Related to Euler's phi-function, January 2009.
Kutztown University
Invited Speaker, Fields containing consecutive elements of order n, Department
Seminar, Spring 2017.
Mid-Atlantic Seminar On Numbers (MASON)
Classifying Super Totient Numbers, MASON IV, Spring 2020.
Odd Coverings of Subsets of the Integers, MASON I, Spring 2018.
Palmetto Number Theory Series (PANTS)
Invited Speaker, Odd Coverings of Subsets of the Integers, PANTS XXXIII, December 2019.
Two Questions Concerning Covering Systems, PANTS XXII, December 2014.
The Reducibility of Constant-Perturbed Products of Cyclotomic Polynomials,
PANTS XX, September 2013.
Factorization of f(x)x^n + g(x) when deg f\leq 2, PANTS XVIII, September 2012.
On a Polynomial Variation of Sierpinski Numbers, PANTS XVI, September 2011.
On the Iteration of a Function Related to Euler’s phi-Function, PANTS VIII, De-
cember 2008.
Shippensburg University
Invited Speaker, Applying a Concept of Pillai to Graph Theory, Department Sem-
inar, Fall 2018.
Invited Speaker, Fields containing consecutive elements of order n, Department
Seminar, Fall 2016.
Invited Speaker, Covering Systems and Some Applications, Department Seminar,
Spring 2013.
Invited Speaker, On the Factorization of the Trinomials x^n+cx^(n-1)+d, Department
Seminar, Spring 2012.
Invited Speaker, On the Iteration of a Function Related to Euler’s phi-Function, Department Semi-
nar, Fall 2008.
SouthEast Regional Meeting On Numbers (SERMON)
On the Factorization of the Trinomials x^n+cx^(n-1)+d, April, 2012.
University of South Carolina
The Factorization of f(x)x^n +g(x) when deg f\leq 2, Number Theory Seminar, Fall
2012.
Covering Systems and Some Applications, SIAM Student Seminar, Fall 2012.
On the Factorization of the Trinomials x^n+cx^(n-1)+d, Number Theory Seminar,
Spring 2012.
On a Polynomial Variation of Sierpinski Numbers, Number Theory Seminar, Fall
2011.
Washington and Lee University
Invited speaker, On the Factorization of the Trinomials x^n+cx^(n-1)+d, Colloquium,
Fall 2012
Selected Service
Co-founder/organizer of the Series on Exploring Combinatorics And Number Theory (SECANT), 2018-2019.
Co-founder and Editor in Chief of Communications on Number Theory and Combinatorial Theory (CONTACT), 2019-2020.
Mentor at Muhlenberg College REU on Challenges in Identifying Integer Sequences, 2016-2018.
Co-PI on NSF REU site grant for Moravian College REU on Research Chellenges of Computational and Experimental Mathematics.
Service at Cedar Crest College.
Faculty secretary, 2016.
Faculty mentor, 2016-2017 AY and 2019-2020 AY.
Adviser for Math Club, 2015-present.
Information Services and Technology Committee, 2016-2017 AY and 2017-2018 AY.
Faculty Development Committee, 2017-2018 AY, 2018-2019 AY, and chaired the committee 2019-2020 AY.
Tenure and Promotion Committee, 2019-2020 AY and 2020-2021 AY.
Teaching at Cedar Crest College
Fall 2014:
MAT 102: College Mathematics
MAT 141: Calculus I
MAT 142: Calculus II
MAT 338: Number Theory
Spring 2015:
MAT 102: College Mathematics
MAT 140: Precalculus
MAT 141: Calculus I
MAT 142: Calculus II
MAT 211: Calculus III
Summer 2015:
MAT 110: Probability and Statistics Online
MAT 110: Probability and Statistics
Fall 2015:
MAT 102: College Mathematics
MAT 110: Probability and Statistics
MAT 142: Calculus II
MAT 311: Linear Algebra
Spring 2016:
MAT 140: Precalculus
MAT 211: Calculus III
HON 324: Honors Exploration in Math and Logic
MAT 316: Modern Algebra
Summer 2016:
MAT 102: College Mathematics
MAT 110: Probability and Statistics Online
MAT 110: Probability and Statistics
Fall 2016:
MAT 102: College Mathematics (two sections)
MAT 140: Precalculus
MAT 141: Calculus I (two sections)
MAT 390: Independent Study in Number Theory
Winter 2016:
MAT 110: Probability and Statistics Online
Spring 2017:
MAT 102: College Mathematics
MAT 110: Probability and Statistics
MAT 141: Calculus I
MAT 142: Calculus II (two sections)
MAT 211: Calculus III
Summer 2017:
MAT 110: Probability and Statistics Online
MAT 110: Probability and Statistics
Fall 2017:
MAT 102: College Mathematics (two sections)
MAT 110: Probability and Statistics
MAT 141: Calculus I
MAT 142: Calculus II
MAT 208: Mathematical Modeling
MAT 311: Linear Algebra
Winter 2017:
MAT 110: Probability and Statistics Online
Spring 2018:
MAT 102: College Mathematics
MAT 110: Probability and Statistics
MAT 140: Precalculus
MAT 141: Calculus I
MAT 211: Calculus III
MAT 338: Number Theory
MAT 316: Modern Algebra
Summer 2018:
MAT 102: College Mathematics
MAT 110: Probability and Statistics Online
MAT 110: Probability and Statistics
Fall 2018:
MAT 107: Mathematics for Healthcare Professionals
MAT 113: ALEKS Prep Lab
MAT 140: Precalculus (two sections)
MAT 141: Calculus I (two sections)
MAT 350: Advanced Calculus
MAT 208: Mathematical Modeling
MAT 360: Selected Topics in Mathematical Research
Winter 2018:
MAT 110: Probability and Statistics Online
Spring 2019:
MAT 110: Probability and Statistics
MAT 141: Calculus I
MAT 142: Calculus II
MAT 380: Independent Research in Mathematics (two students)
MAT 390: Independent Study in Algebraic Number Theory
Summer 2019:
MAT 102: College Mathematics
MAT 110: Probability and Statistics Online
MAT 110: Probability and Statistics (two sections)
Fall 2019:
MAT 113: ALEKS Prep Lab
MAT 140: Precalculus
MAT 141: Calculus I
MAT 142: Calculus II
MAT 311: Linear Algebra
MAT 360: Selected Topics in the Theory of Irreducible Polynomials
MAT 380: Independent Research in Mathematics (three students)
Winter 2019:
MAT 110: Probability and Statistics Online
Spring 2020:
MAT 102: College Mathematics
MAT 141: Calculus I
MAT 208: Mathematical Modeling
MAT 211: Calculus III
MAT 224: Discrete Mathematics
MAT 316: Modern Algebra
MAT 380: Independent Research in Mathematics (three students)
Summer 2020:
MAT 110: Probability and Statistics Online (two sections)
Fall 2020:
MAT 110: Probability and Statistics (two sections, hybrid)
MAT 140: Precalculus (hybrid)
MAT 141: Calculus (two section, hybrid)
MAT 208: Mathematical Modeling (hybrid)
MAT 380: Independent Research in Mathematics (three students)
MAT 113: ALEKS Prep Lab
Winter 2020:
MAT 110: Probability and Statistics Online
Spring 2021:
MAT 110: Probability and Statistics (three sections, online)
MAT 141: Calculus (online)
MAT 142: Calculus II (hybrid)
MAT 211: Calculus III (hybrid)
MAT 380: Independent Research in Mathematics (four students)