## Joshua HarringtonAssociate Professor of MathematicsCurtis 221Chair of the Mathematics Department Cedar Crest College Office:
Email: Joshua.Harrington@cedarcrest.edu |

**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.**Two questions concerning covering systems**, Int. J. Number Theory 11, 2015, 1739-1750.**Special numbers in the ring Zn**with**Samuel Gross**, J. Integer Seq. 18, 2015, no 11, Article 15.11.3.**A problem of Diophantus modulo a prime**with**Lenny Jones**, Irish Math. Soc. Bull. no. 77 (2016), 45-49.**Extending some irreducibility results of Finch and Jones**with**Lenny Jones**, J. Comb. Number Theory 8 (2016), no. 2, 131-144.**On the congruence x^x mod n**with**James Hammer**and**Lenny Jones**, Integers 16, 2016.**The irreducibility of power compositional polynomials and their Galois groups**with**Lenny Jones**, Math. Scand. 120 (2017), no. 2, 5-16.**New primitive covering numbers and their properties**with**Lenny Jones**and**Tristan Phillips**, J. Number Theory 172, 2017, 160-177**On consecutive primitive nth roots of unity modulo q**with**Thomas Brazelton**,**Siddarth Kannan**, and**Matthew Litman**, J. Number Theory 174, 2017, 495-504.**Sums of polynomial residues**with**Samuel Gross**and**Laurel Minott**, Irish Math. Soc. Bull. no. 79 (2017), 31-37.**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**.**Monogenic cyclotomic composotions**, with**Lenny Jones**.**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**.**Two dependent probabilistic chip-collecting games**with**Kedar Karhadkar**,**Madeline Kohutka**,**Tessa Stevens**, and**Tony W. H. Wong**.**Monogenic binomial compositions**, 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. Grotth**and**Carrie Finch-Smith.****Some new polynomial discriminants**, with**Lenny Jones.****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.****On k-fold super totient numbers**, with**Melea Roman**and**Tony W. H. Wong**.

- 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**.

- 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 Sem- inar, 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

- 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.

- 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