Joshua Harrington

Associate Professor of Mathematics
Chair of the Mathematics Department
Cedar Crest College
Office:
Curtis 221
Email:
 Joshua.Harrington@cedarcrest.edu

Published Research

  1. On the iteration of a function related to Euler's phi-function with Lenny Jones, Integers 10, 2010.
  2. On the Factorization of the Trinomials x^n+cx^(n-1)+d, Int. J. Number Theory 8, 2012, 1513-1518.
  3. Arithmetic progressions in the polygonal numbers with Scott Dunn and Kenneth Brown, Integers 12, 2012.
  4. A polynomial investigation inspired by work of Schinzel and Sierpinski with Michael Filaseta, Acta Arith. 155, 2012, 149-161.
  5. Nonlinear Sierpinski and Riesel Numbers with Carrie Finch and Lenny Jones, J. Number Theory 133, 2013, 534-544.
  6. A class of irreducible polynomials with Lenny Jones , Colloq. Math. 132, 2013, 113-119.
  7. The reducibility of constant-perturbed products of cyclotomic polynomials with Lenny Jones and Daniel White, Int. J. Number Theory 10, 2013, 13-29.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. Extending a theorem of Pillai to quadratic sequences with Lenny Jones, INTEGERS 15A.
  13. Two questions concerning covering systems, Int. J. Number Theory 11, 2015, 1739-1750.
  14. Special numbers in the ring Zn with Samuel Gross, J. Integer Seq. 18, 2015, no 11, Article 15.11.3.
  15. A problem of Diophantus modulo a prime with Lenny Jones, Irish Math. Soc. Bull. no. 77 (2016), 45-49.
  16. Extending some irreducibility results of Finch and Jones with Lenny Jones, J. Comb. Number Theory 8 (2016), no. 2, 131-144.
  17. On the congruence x^x mod n with James Hammer and Lenny Jones, Integers 16, 2016.
  18. The irreducibility of power compositional polynomials and their Galois groups with Lenny Jones , Math. Scand. 120 (2017), no. 2, 5-16.
  19. New primitive covering numbers and their properties with Lenny Jones and Tristan Phillips, J. Number Theory 172, 2017, 160-177
  20. On consecutive primitive nth roots of unity modulo q with Thomas Brazelton, Siddarth Kannan, and Matthew Litman, J. Number Theory 174, 2017, 495-504.
  21. Sums of polynomial residues with Samuel Gross and Laurel Minott, Irish Math. Soc. Bull. no. 79 (2017), 31-37.
  22. Differences Between Elements of the Same Order in a Finite Field, with Lenny Jones, J. Number Theory 180, 2017, 443-459
  23. An equation involving arithmetic functions and Riesel numbers, with Lenny Jones, Integers 18, 2018.
  24. A new condition equivalent to the Ankeny-Artin-Chowla conjecture, with Lenny Jones, J. Number Theory 192, 2018, 240-250.
  25. A modification of a problem of Diophantus, with Lenny Jones, Math. Slovaca 68, no. 6, 2018, 1343-1352.
  26. Odd coverings of subsets of the integers, with James Hammer and Kristina Marotta, J. Comb. Number Theory, 2018, Volume 10, Issue 2.

  27. 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.
  28. On super totient numbers and super totient labelings of graphs, with Tony W.H. Wong, Discrete Math. 343 (2020), no. 2, 111670, 11 pp.

Submitted Research

  1. Graph polynomials for a class of DI-pathological graphs, with James Hammer.

  2. Monogenic cyclotomic composotions, with Lenny Jones.

  3. 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.
  4. Two dependent probabilistic chip-collecting games with Kedar Karhadkar, Madeline Kohutka, Tessa Stevens, and Tony W. H. Wong.
  5. Monogenic binomial compositions, with Lenny Jones.
  6. Sums of distinct cubic polynomial residues, with Carrie Finch-Smith and Tony W. H. Wong.
  7. Sum index and difference index of graphs, with Eugene Henninger-Voss, Kedar Karhadkar, Emily Robinson, and Tony W. H. Wong.
  8. Consecutive Sierpinski numbers, with R. Grotth and Carrie Finch-Smith.
  9. Some new polynomial discriminants, with Lenny Jones.
  10. 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.
  11. On k-fold super totient numbers, with Melea Roman and Tony W. H. Wong.

Contributions to the Online Encyclopedia of Integer Sequences

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.
  7. 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.
  8. 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.
  9. 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.

Presentations

Selected Service

Teaching at Cedar Crest College