A valuable memory

This paper by the legendary N. J. A. Sloane mentioning the result on prime sequences (particularly GPF-Fibonacci sequences and a conjecture on GPF-Tribonacci) that I got in collaboration with my student Greg Back meant a lot to me. It is, for me, a most valuable "Math memory".

 

The Phase Transition in Random Graphs: A Simple Proof

 

CV - Mihai Caragiu (12/3/2025)

M

 

m-caragiu.1@onu.edu Office: 419-772-2352  February 7, 2026

 

BIOGRAPHY

• 8/11 - present: Professor of Mathematics, Ohio Northern University
• 9/04 - 8/11: Associate Professor of Mathematics, Ohio Northern University (tenured since 8/06) 
• 8/00 - 9/04: Assistant Professor of Mathematics, Ohio Northern University
• 7/99 - 7/00: Research Associate, Education Program for Gifted Youth, Stanford University
• 8/96 - 5/99: Assistant Professor, Department of Pure and Applied Mathematics, Washington State University
•  8/92 - 8/96: Graduate Teaching Assistant, Department of Mathematics, Pennsylvania State University

EDUCATION

• B. Sc. (1987) University of Bucharest
• M. Sc. (1988) University of Bucharest
• Ph. D. (1996) The Pennsylvania State University

TEACHING

Courses taught at ONU include: Discrete Mathematics, Linear Algebra, Abstract Algebra, Foundations of Mathematics, Differential Equations, Calculus for Engineers, Calculus for Life Sciences, College Algebra, Number Theory and Cryptography, Probability and Graphs, Junior Seminar, Sophomore Seminar, Freshman Seminar, Mathematical Problem Solving, Senior Capstone.

AWARDS

• Fall Semester 2022-2023: sabbatical leave awarded
• 2021-2022 Mary Reichelderfer Chair of Mathematics, Ohio Northern University
• 2018 Faculty Research Award, the Getty College of Arts and Sciences (the first such award)
• 2017 Outstanding Teaching Faculty Award for Mathematics and Statistics, Ohio Northern University
• 2016-2017 Mary Reichelderfer Chair of Mathematics, Ohio Northern University
• Summer Research Stipend, Ohio Northern University, 2015
• Top 25 STEM Professors in Ohio (2013)
• 2011-2012 Mary Reichelderfer Chair of Mathematics, Ohio Northern University
• 2011 Outstanding Teaching Faculty Award for Mathematics and Statistics, Ohio Northern University 
• 2010 Outstanding Teaching Faculty Award for Mathematics and Statistics, Ohio Northern University
• Summer Research Stipend, Ohio Northern University, 2010
• Winter Quarter 2008-2009: sabbatical leave awarded
• 2007-2008 Mary Reichelderfer Chair of Mathematics, Ohio Northern University
• 2004-2005 Mary Reichelderfer Chair of Mathematics, Ohio Northern University
• Pritchard Dissertation Fellowship, Penn State, 1996
• Wheeler P. Davey Memorial Scholarship, Penn State, 1995
• Wollmer-Klechner Scholarship in Science, Penn State, 1993 
• Traian Lalescu awards, mathematics undergraduate contests, 1983 and 1984
• Member of the Mathematics Olympiad Team, Romania 1979-1982

RESEARCH INTERESTS

• Experimental Mathematics. Integer sequences. Prime Numbers. Greatest Prime Factor Sequences.
• Elementary and analytic number theory and their Applications. Cryptography.
• Fibonacci numbers. Ducci games and their analogues.
• Random Structures. Mathematical Physics.
• Mathematics Education. Undergraduate Research.

PEER REVIEWED PUBLICATIONS

1. Mihai Caragiu. Proportion Patterns with Binomials. Far East J. of Math. Education 28(1), 28-37 (2006) - https://doi.org/10.17654/0973563126005 
2. Mihai Caragiu and Dimitar Bakalov. In memoriam: Khristo Nonev Boyadzhiev. Journal of Geometry and Symmetry in Physics 73, 63-73 (2025).
3. Mihai Caragiu and Mellita Caragiu. A Random Variable with Zeta Function Connections. Far East J. of Math. Education, Vol. 27, no.1, 21-33 (February 2025)
4. Mihai Caragiu and Kaleb Swieringa. On the alternating sum-of-divisors. JP Journal of Algebra Number Theory and Appl. 63, No. 2, 97-110 (2024). Journal publication with an ONU student co-author (Kaleb Swieringa).
5. Mihai Caragiu and Rachael Harbaugh. Extending a Putnam Problem to Fields of Various Characteristics. JP Journal of Algebra, Number Theory, and Appl. Vol 59, 33-45 (November 2022). Journal publication with an ONU student co-author (Rachael Harbaugh).
6. Mihai Caragiu, An Elementary Note on the Greatest Prime Factors of Linearly Related Integers, JP Journal of Algebra, Number Theory, and Appl. Vol 52(1), 95 - 100 (October 2021), http://dx.doi.org/10.17654/NT052010095
7. Mihai  Caragiu and Addison Carter, Random Compositions for the Undergraduate Classroom. Far East J. of Math. Education Volume 19, Issue 2, Pages 129 - 139 (June 2019). Journal publication with an ONU student co-author (Addison Carter).
8. Mihai Caragiu, Shannon Tefft, Aaron Kemats and Travis Maenle. A linear complexity analysis of quadratic residues and primitive roots spacings. Far East J. of Math. Education Volume 19, Issue 1, Pages 27 - 37 (February 2019), https://arxiv.org/abs/1902.07314 Journal publication with three ONU student co-authors (Shannon Tefft, Aaron Kemats and Travis Maenle).
9. Mihai Caragiu. Sequential experiments with primes. Cham: Springer (ISBN 978-3-319-56761-7/hbk; 978-3-319- 56762-4/ebook). xi, 279 p. (2017). Springer research monograph (featuring several undergraduate research themes) - https://link.springer.com/book/10.1007/978-3-319-56762-4
10. Mihai Caragiu, Paul A. Vicol, and Mohammad Zaki. On Conway’s subprime function, a covering of ℕ and an unexpected appearance of the Golden ratio. Fibonacci Quarterly 55, No. 4, 327-331 (2017).
11. Mihai Caragiu and Paul A. Vicol. Prime magmas and a cyclicity conjecture. JP Journal of Algebra, Number Theory and Appl. 38, No. 2, 129-143 (2016).
12. Mihai Caragiu, Alexandru Zaharescu and Mohammad Zaki. On Ducci sequences with primes. Fibonacci Quarterly 52, No. 1, 32-38 (2014).
13. Mihai Caragiu, Donald Pleshinger and Jonathan Schroeder. Uniform distribution for a class of k-paradoxical oriented graphs. JP Journal of Algebra, Number Theory and Appl. 29, No. 2, 107-117 (2013). Journal publication with two ONU student co-authors (Donald Pleshinger and Jonathan Schroeder).
14. Mihai Caragiu, Alexandru Zaharescu and Mohammad Zaki. An analogue of the Proth-Gilbreath conjecture. Far East J. Math. Sci. (FJMS) 81, No. 1, 1-12 (2013).
15. Mihai Caragiu and Courtney Brown. Quadratic residues and a special class of polynomials. Far East J. Math. Educ. 8, No. 1, 43-50 (2012). Journal publication with an ONU student co-author (Courtney Brown).
16. Mihai Caragiu, Alexandru Zaharescu and Mohammad Zaki. On a class of solvable recurrences with primes. JP Journal of Algebra, Number Theory and Appl. 26, No. 2, 197-208 (2012).
17. Mihai Caragiu, Alexandru Zaharescu and Mohammad Zaki. On Ducci sequences with algebraic numbers. Fibonacci Quarterly 49, No. 1, 34-40 (2011).
18. Mihai Caragiu, Mohammad Zaki and Lauren Sutherland. Multidimensional greatest prime factor sequences. JP Journal of Algebra, Number Theory and Appl. 23, No. 2, 187-195 (2011). Journal publication with an ONU student co-author (Lauren Sutherland).
19. Mihai Caragiu and Ashley Risch. An Euler-Fibonacci sequence. Far East J. Math. Sci. (FJMS) 52, No. 1, 1-7 (2011). Journal publication with an ONU student co-author (Ashley Risch).
20. Mihai Caragiu. Continuously composed rotations. Far East Journal of Mathematical Sciences, 39(2), 261-266 (April 2010).
21. Mihai Caragiu. On an inequality proposed by A. Lupas. Far East J. Math. Educ. 4, No. 1, 11-14 (2010).
22. Greg Back and Mihai Caragiu. The greatest prime factor and recurrent sequences. Fibonacci Quarterly 48, No. 4, 358-362 (2010). Journal publication with an ONU student co-author (Greg Back).
23. Mihai Caragiu. Recurrences based on the greatest prime factor function. JP Journal of Algebra, Number Theory and Appl. 19, No. 2, 155-163 (2010).
24. Mihai Caragiu and Greg Back. The greatest prime factor and related magmas. JP Journal of Algebra, Number Theory and Appl. 15, No. 2, 127-136 (2009). Journal publication with an ONU student co-author (Greg Back).
25. Mihai Caragiu, Ronald Johns and Sandra Schroeder. On the combined use of algebra and technology in the study of a family of sequences. Far East J. Math. Educ. 3, No. 1, 99-104 (2009).
26. Florin Caragiu and Mihai Caragiu. Discrete Structures as Holistic Models. Transdisciplinarity in Science and Religion No. 3, 103-110 (2008).
27. Mihai Caragiu and John Holodnak. On sampling periodic functions. Far East J. Math. Sci. (FJMS) 29, No. 1, 145- 149 (2008). Journal publication with an ONU student co-author (John Holodnak).
28. Mihai Caragiu. Discrete Fourier transforms and plane rotations. Adv. Appl. Discrete Math. 2, No. 2, 151-157 (2008).
29. Mihai Caragiu and Nathan Baxter. A note on p-adic Ducci games. JP Journal of Algebra, Number Theory and Appl. 8, No. 1, 115-120 (2007). Journal publication with an ONU student co-author (Nathan Baxter).
30.  Mihai Caragiu and Lisa Scheckelhoff. The greatest prime factor and related sequences. JP Journal of Algebra, Number Theory and Appl. 6, No. 2, 403-409 (2006). Journal publication with an ONU student co-author (Lisa Scheckelhoff).
31. Mihai Caragiu and Laurence Robinson. An intermediate value theorem for sequences with terms in a finite set. JP Journal of Algebra, Number Theory and Appl. 6, No. 1, 57-70 (2006).
32. Mihai Caragiu, Ronald Johns and Justin Gieseler. Quasi-random structures from elliptic curves. JP Journal of Algebra, Number Theory and Appl. 6, No. 3, 561-571 (2006). Journal publication with an ONU student co-author (Justin Gieseler).
33. Mihai Caragiu. Codekets. Far East J. Math. Sci. (FJMS) 21, No. 2, 133-141 (2006).
34. Mihai Caragiu. A note on codes and kets. Sibirskie Èlektronnye Matematicheskie Izvestiya 2, 79-82 (2005).
35. Florin Caragiu and Mihai Caragiu. On the ranges of discrete exponentials. Int. J. Math. Math. Sci. 2004, No. 41- 44, 2265-2268 (2004).
36. Nathan Baxter and Mihai Caragiu. Arithmetic properties of some special sums. JP Journal of Algebra, Number Theory and Appl. 4, No. 3, 455-463 (2004). This was the first Mathematics journal publication with an ONU student co-author (Nathan Baxter).
37. Mihai Caragiu and William Webb. On modular Fibonacci sets. Fibonacci Quarterly 41, No. 4, 307-309 (2003).
38. Mihai Caragiu. Zero sets of polynomials: one versus two variables. Elemente der Mathematik 57, No. 2, 76-79 (2002).
39. Mihai Caragiu. Constructing irreducible polynomials with prescribed level curves over finite fields. Int. J. Math. Math. Sci. 27, No. 4, 197-200 (2001).
40. Mihai Caragiu. Multivariate interpolation by absolutely irreducible polynomials over finite fields. Revue Roumaine de Mathématiques Pures et Appliquées 46, No. 6, 719-724 (2001).
41. Mihai Caragiu and William Webb. Invariants for linear recurrences. Howard, Fredric T. (ed.), Appl. of Fibonacci numbers. Volume 8: Proceedings of the eighth international research conference on Fibonacci numbers and their Appl., Rochester, NY, USA, June 22-26, 1998.
42. Mihai Caragiu. First-order non-definability for primitive roots. Revue Roumaine de Mathématiques Pures et Appliquées 44, No. 2, 167-169 (1999).
43. Mihai Caragiu and Mellita Caragiu. The GL (n, p) – invariance of the Potts Hamiltonian. Int. J. Math. Math. Sci. 20, No. 1, 33-36 (1997).
44. Mihai Caragiu and Gary L. Mullen. The distribution of power residues in finite fields. Southeast Asian Bulletin of Mathematics 21, No. 2, 149-157 (1997).
45. Mihai Caragiu. On a class of constant weight codes. Electronic Journal of Combinatorics 3, No. 1, Research paper R4, 13 p. (1996); printed version J. Comb. 3, No. 1, 43-55 (1996).
46. Mihai Caragiu. On a class of finite upper half-planes. Discrete Mathematics 162, No. 1-3, 49-66 (1996).
47. Mihai Caragiu. On the combinatorics of squares in some zero characteristic fields. Nieuw Archief voor Wiskunde. Vierde Serie, IV. Ser. 13, No. 2, 209-218 (1995).
48. Mihai Caragiu and Mellita Vicol. A field with an unusual square distribution. Mathematica 36(59), No. 1, 21-23 (1994).
49. Mihai Caragiu. Counting the maximal sequences of consecutive quadratic residues modulo p. Revue Roumaine de Mathématiques Pures et Appliquées 38, No. 9, 745-749 (1993).
50. Mihai Caragiu and Mellita Vicol. Ternary quadratic forms over differential fields, with Appl. in physics. Mathematica 35(58), No. 2, 123-126 (1993).
51. Mihai Caragiu. Representations of groups through automorphisms of generalized metric spaces. Stud. Cercet. Mat. 44, No. 4, 285-288 (1992).
52. Mihai Caragiu. The Riemann hypothesis and the logic of finite fields. Proceedings of the 9th national conference on algebra held at the University of Cluj-Napoca, Romania, September 18-20, 1991.
53. Mihai Caragiu. Equations in Abelian groups. Gazeta Matematica, Perfecţ. Metod. Metodol. Mat. Inf. 13, No. 1-2, 41-47 (1992).

PROBLEM – POSING ACTIVITY

1.     Mihai Caragiu. Problem 2021 – Mathematics Magazine (June 2017).

BOOK REVIEWS NON-FICTION

1.     Mihai Caragiu. David Berlinski: "One, Two, Three: Absolutely Elementary Mathematics" (Pantheon Books, 2011) book review published in the Journal of the ACMS (September 2011).

MATHEMATICS TEXTBOOKS REVIEWING

1.     Reviewed a Discrete Mathematics book proposal for Birkhäuser/Springer (March 2022)

CONFERENCE PRESENTATIONS
(INT – International, INV – Invited, SPEC – Special Session Talk)

1. 2021 Fall Meeting of the Ohio MAA, University of Toledo. Title: “Being an experimentalist: classroom explorations in Experimental Math”
2. 2017 Fall Meeting of the Ohio MAA invited address, Ohio University Eastern. Title: “Sequential Experiments with Primes” (INV)
3. 2016 Math Fest, Columbus OH. Special Session on Programming in Mathematics Classes and Mathematics for Programming. Title: “Computational Number Theory - Quest and Discovery in the Undergraduate Classroom” (SPEC)
4. 2016 Ohio MAA Spring Meeting, Ohio Northern University. Title: “Romanian Mathematics Baccalaureate Exam (2015)”
5. 2015 Ohio MAA Fall Meeting, Capital University. Title: “All primes in terms of one: non-associative algebra and Google cloud computing”
6. 2013 Miami University OH Fall Conference. Title: “Making it count: undergraduate research with an impact”.
7. Mihai Caragiu. Uniform distribution for a class of k-paradoxical oriented graphs. Special Session on Discrete Mathematics and Theoretical Computer Science, Joint International Meeting of the American Mathematical Society and the Romanian Mathematical Society, June 27 - 30, 2013, Alba Iulia, Romania (INT, SPEC)
8. 2012 University of Findlay’s Mathematics Colloquium invited talk. Title: “Beyond High School Science Fairs: The Senior Capstone Project” (INV)
9. 2009 Miami University, OH - 9/26. Conference on the Teaching of Undergraduate Mathematics. Title: "Difference Quotient Revisited".
10. 2009 Ohio MAA Spring Meeting, Bowling Green State University, April 3. Title: Rotations and translations revisited.
11. 2009 Joint Mathematics Meetings, Washington D.C., January 6: Sandra Schroeder (presenter) and Mihai Caragiu "On the combined use of algebra and technology in the study of a family of sequences".
12. 2008 Recreational Mathematics Conference, Miami University, OH, September 27. Title: "Computer Art with Elliptic Curves".
13. 2007 Ohio MAA Fall Meeting, Wittenberg University. Invited address: "Geometry with Complex Numbers". (INV)
14. 2007 Number Theory Conference, Miami University, Oxford, OH, September 28. Title: "Ultimate periodicity for a special class of GPF sequences"
15. 2007 Ohio MAA Spring Meeting, Shawnee State University. Title: "Congruential Extensions of Ducci Games".
16. 2007 AMS Spring Central Section Meeting, Miami University, Oxford OH, March 16. Title: On p-adic Ducci Games
17. 2006 AMS Fall Central Section Meeting, University of Cincinnati, October 21. Title: Recurrent sequences based on the greatest prime factor function
18. 2005 Ohio MAA Spring Meeting, Miami University. Title: "Small orders modulo p".
19. March 3, 2005 - Invited speaker in Penn State's Algebra and Number Theory Seminar. Title: "Some results involving sequences and graphs". (INV)
20. Mathematics and Symmetry Conference, Miami University, Oct 2-3, 2004. Title: "Quadratic residues between symmetry and randomness".
21. Special session on Fibonacci Numbers, 2004 Ohio MAA Spring Meeting, Cincinnati: "Entangled Lucas Numbers" (SPEC)
22. 2003 Ohio MAA Fall Meeting, Ohio Northern University. Title: "What is a spooky slice from a Lucas number?"
23. Discrete Mathematics & Its Appl. Conference, Miami University, October 3-4, 2003. Title: "Building bridges towards Physics in the Discrete Mathematics Class" (with Mellita Caragiu).
24. 2003 Ohio MAA Spring Meeting, Ohio State University. Title: "Cassini identity: a combinatorial proof".
25. 2002 Ohio MAA Spring Meeting, Xavier University. Title: "A new way of looking at primitive roots".
26. 2001 Ohio MAA Fall Meeting, Marietta College. Title:  "Power Residues and Residual Randomness".
27. 2001 Ohio MAA Spring Meeting, Bowling Green University. Title: "The parity-check code viewed as an Ising model".
28. Miami University Math Conference, September 2000. Title: "Multivariate interpolation over Galois fields".
29. William Webb (presenter) and Mihai Caragiu, 9th International Conference on Fibonacci Numbers and Their Appl., Luxembourg, July 2000. Title: "Homogeneous Polynomial Identities for k-th Order Recurrences". (INT)
30. Spring 2000: University of Missouri, Columbia. Title: "Linear Recurrences Revisited".
31. Summer 1999: Stanford University, (Stanford Mathematics Camp Guest Lecturer). Title: "Cellular Automata".
32. Fall 1997: The University of Montana, Missoula. Title: "Finite Fields, Codes and Quasi-randomness".
33. Spring 1997: Washington State University. Title: "First-order non-definability of primitive roots".
34. Fall 1996: Washington State University. Title: "Partitions, q-series and... Fermions (on a result of Richard Borcherds)".
35. Spring 1996: Washington State University. Title: "On a class of nonlinear codes".

OHIO NORTHERN MATHEMATICS SEMINAR PRESENTATIONS

1. Fall 2022. “The Dirac’s Delta function”
2. Fall 2022. “The 2N+1 Problem”
3. Fall 2017. “The top problem in the MAA Ohio Spring Competition: a discussion”
4. Fall 2016. “Computational number theory - quest and discovery in the undergraduate classroom”
5. Spring 2016. “Publications: exploring the mathematical Pale Blue Dot”
6. Fall 2015. “Experimental Mathematics: Primes, Sequences, and Non-Associative Algebra”
7. Spring 2009. “Special Classes of Integer Sequences”
8. Fall 2008. “Designing Nice Rugs by Using Number Theory”
9. Fall 2007. “Trapping Prime Sequences”
10. Fall 2007. “Simson lines and Nine-Point Circles”
11. Fall 2006. “The 2006 Putnam Exam”
12. Spring 2006. “Tridents, Thumbtacks, and Point Set Topology”
13. Fall 2005. “Some Interesting Induction Problems
14. Fall 2004. “Quadratic residues between symmetry and randomness”
15. Fall 2003. "p-adic Numbers" (parts 1 and 2)
16. Fall 2002. "Topological Quantum Field Theory: An Introduction" (parts 1 and 2)
17. Fall 2001. "Beam me up! Quantum nonlocality, Pauli matrices, Codes and all that..."
18. Fall 2001. "Primes, Polynomials and Modular Sequences".
19. Winter 2000. "Extension of Algebra Operations: Category Theory"

OUTREACH (HIGH SCHOOL/MIDDLE SCHOOL)

1. Hyperbolic geometry (First Annual Math Awareness Day, Spring 2002, ONU)
2. A journey through Cryptography (Third Annual Math Awareness Day, Spring 2004, ONU)
3. Number Games (workshop given at the Fourth Annual Math Awareness Day, Spring 2005, ONU)
4. July 10, 2007. ONU "MI READY" Workshop: Number Theory and Cryptography
5. Geometrical Transformations with Complex Numbers (Summer Honors Institute, ONU, 2006)
6. Geometrical Transformations with Complex Numbers (Summer Honors Institute, ONU, June 2007)
7. Cryptography (Summer Honors Institute, ONU 2013, 2014, 2015, 2016, 2018.

UNDERGRADUATE RESEARCH

37 senior research (capstone) projects advised since 2003
15 journal articles published with undergraduate co-authors
40 conference presentations by student advisees, most ever for an Ohio Northern Math/Stat faculty (first one, at the 2001 MAA Ohio Spring Meeting in Bowling Green, initiated the streak of 107 ONU student presentations at mathematical meetings since 2001).


CAPSTONE PROJECTS

1. Kaleb Swieringa, On the Alternating Sum of Divisors 2023
2. McKinley Britton: Fibonacci Numbers and Domino Tilings 2023
3. Alexander Hare: Experiments with Greatest Prime Factor Sequences 2023
4. Rachael Harbaugh: Extending a Putnam Problem to Fields of Various Characteristics 2022
5. Benjamin Morris: Fibonacci Periods 2022
6. Greg Hassenpflug: The Golden Ratio 2021
7. Aaron Kemats: An Investigation of the Square Grid Graph 2021
8. Travis Maenle: A linear complexity analysis of quadratic residues and primitive roots spacings 2020
9. Bryan Peck: Bell's Inequalities 2020
10. Kaity Kelly: Gaussian Integers 2020
11. Kenneth Eaton: The Fundamentals of Automated Theorem Proving 2019
12. Megan Meyer: An Experimental Approach to Sophie Germain Sequences 2019
13. Addison Carter: An Introduction to Partitions and Compositions 2019
14. Rachel Liebrecht: Special Topics on Graph Theory and Ramsey Numbers 2019
15. Shannon Tefft: Processing Quadratic Residues with Ducci Iterations 2019
16. Jenna Holler: American Mathematics Competitions – Variations and Generalizations 2018
17. Joseph Stomps: American Mathematics Competitions AMC 10 - analogies and generalizations 2018
18. Matthew Golden: The Baker-Campbell-Hausdorff Formula 2017
19. Michelle Haver: Poissonian Character and Chebyshev Bias for GPF Sequences: A Computational Analysis 2017
20. Amanda Marco: Fibonacci Numbers and Some of Their Properties 2014
21. Matthew R. Zirkle: Finding Square Roots in a Prime Field 2013
22. Jonathan C. Schroeder: Small Special Pairs of Primitive Roots 2013
23. Donald J. Pleshinger: On a Congruence of Ohtsuka 2013
24. Ashley Risch: An Euler-Fibonacci Sequence 2011
25. Lauren Sutherland: Multidimensional Greatest Prime Factor Sequences 2011
26. Greg Back: The Greatest Prime Factor and its Applications 2010
27. Jenna Brace: Traffic Flow Simulation with Cellular Automata 2010
28. John Holodnak: The Perron-Frobenius Theorem and Applications 2010
29. Sharon Binkley: The One Time Pad and Text Visualization 2009
30. Joshua Somerlot: The Affine Cipher 2009
31. Andrew Homan: An Overview of Model Theory and Completeness 2007
32. Allison Mackay: Elementary Number Theory and Classical Cryptography 2006
33. Lisa Scheckelhoff: GPF Sequences 2006
34. Brandon Bucholtz: The Euclidean Algorithm 2006
35. Jacob L. Johanssen: Fibonacci-Lucas Densities 2006
36. Nathan Baxter: Finite Fields 2005
37. Sara Miller: Fibonacci Numbers 2003

LATEST PRESENTATIONS BY ADVISED STUDENTS

1. Kaleb Swieringa (corresponding author), Joelena Brown, Rachael Harbaugh, and Francis Nadolny: Artsy Chaos: The Secret Life of a Class of Trigonometric Sums, 2022 Student Research Colloquium, ONU
2. Alexander Hare: A Strange Attractor with Prime, 2022 Student Research Colloquium, ONU
3. Aaron Kemats and Travis Maenle: "Linear Complexities of Quadratic Residues and Primitive Roots Spacings"- 2018 Ohio MAA Fall Meeting, Malone University, October 27, 2018
4. Shannon Tefft: "Processing Quadratic Residues with Ducci Iterations" - 2018 Ohio MAA Fall Meeting, Malone University, October 27, 2018
5. Aaron Kemats: A Fibonacci-Lucas experiment. Fall Meeting of the Ohio Section of the MAA, Ohio University Eastern, October 28, 2017.
6. Takumi Kijima: Naïve Bayes Classifier. Fall Meeting of the Ohio Section of the MAA, Ohio University Eastern, October 28, 2017.
7. Michelle Haver. On the R. Lemke Oliver - K. Soundararajan recent "prime conspiracy". The Spring Meeting of the Ohio Section of the MAA (“Centennial Meeting”), Ohio Northern University, April 9, 2016.

 
ADVISING AWARD-WINNING STUDENT PRESENTATIONS AT NATIONAL MEETINGS

1. Lisa Scheckelhoff (2007): On a class of recurrent sequences based on the greatest prime factor function.
2. Andrew J. Homan (2007): Robinson's theorem in connection with a Putnam problem.
3. Justin Gieseler (2007): An application of elliptic curves to one-time pad cryptography - project jointly advised by Mihai Caragiu and Ronald Johns.

THE ONU - SOLVE PROBLEM GROUP

In January 2011, together with Dr. Chowdhury we founded the ONU-Solve Problem Group with the purpose of engaging students in solving problems proposed in mainstream math journals with problem sections. The ONU-Solve achievements include 23 mentions of ONU-SOLVE (or ONU-SOLVE members) among those submitting correct solutions (with 3 of those published as featured solutions).

Service and administrative work
 

• Mathematics Program Lead (2020-2022)
• Mathematics Co-Chair, Department of Mathematics and Statistics, Ohio Northern University (2018-2020)
• Chair, of Department of Mathematics and Statistics, Ohio Northern University (2014 – 2018)
• Member in the Getty College of Arts and Sciences’ Council of Department Chairs (CDC) 2014-2018
• Worked on the Mathematics and Statistics website as the departmental web person.
• Numerous meetings with prospecting students
• Initiated a Modern Mathematics cultural literacy poster project for ONU students.
• Organized a local test center for the AMC 10/12 American Mathematical Competitions at Ohio Northern University, for students in neighboring high schools. Two such competitions were held so far (2018, 2019).
• Mentored incoming faculty
• Inviting external speakers in the ONU Math Seminar

Vladimir Arnold - Hilbert 13

Vladimir I. Arnold 

 

On the functions of three variables. Doklady AN USSR, 1957, 114:4, 679-681

Found it in English, among the classic papers collected by Yaoliang Yu @ https://cs.uwaterloo.ca/~y328yu/classics/Arnold57.pdf

A Deterministic Sample of Bourgain’s Work - Peter Sarnak and Avi Wigderson

 

Diagonalization chalkboards (lecture March 19)

 

 

A short classroom exercise on partially ordered sets

 

Noncommutative probability for computer scientists - Adam Marcus

 

Quanta Magazine: The Year in Math

"Landmark results in geometry and number theory marked an exciting year for mathematics, at a time when advances in artificial intelligence are starting to transform the subject’s future."

Full article at  

https://www.quantamagazine.org/the-year-in-math-20241216/

Maze generation with Kruskal's algorithm

 

Flurries of Ducci waves

https://ssmr.ro/bulletin/pdf/66-2/articol_4.pdf

 

 

On the Liouville function at polynomial arguments - Joni Teräväinen

 

Integer partitions detect the primes

A remarkable new paper:

Integer partitions detect the primes, by William Craig, Jan-Willem van Ittersum, Ken Ono

https://arxiv.org/abs/2405.06451

"Model Theory (Ax's Theorem)" by Piotr Kowalski

 

The Riemann Hypothesis, Explained - Quanta Magazine

 

The Geometric Langlands Conjecture - Sam Raskin

 

Sir Timothy Gowers: What is Mathematics? with G-Research

 

The case for 'math-ish' thinking

The case for ‘math-ish’ thinking 

May 15th, 2024 - Stanford Report

In a new book, Jo Boaler argues for a more flexible, creative approach to math. “Stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.”

Paper on subprime tribonacci sequences

 "Three’s a crowd: an exploration of subprime tribonacci sequences" - College Mathematics Journal 54, No. 5, 464-475 (2023) - by Sara Barrows, Emily Noye, Sarah Uttormark, and Matthew Wright.

 

Iwasawa decomposition in a nutshell

 A wonderful article by Laurie Wurth-Pressel (source)

130 years of mind games and quantum challenges

January 22, 2024

 

2024 marks the 130th anniversary of The American Mathematical Monthly. 

The trailblazer who launched this premier academic journal in 1894 and inspired the establishment of the American Mathematical Association (AMA) graduated from Ohio Northern University in the late 1800s—Benjamin Franklin Finkel, BS 1888, BA 1896.

Mathematical problems fascinated Finkel from a young age, and his eagerness to solve them impelled his ambition to “publish a journal devoted solely to mathematics and suitable to the needs of teachers of mathematics.”

He once stated: “Many dormant minds have been aroused into activity through the mastery of a single problem.” 

Over a century later, Finkel’s statement still rings true. 

While much has changed at Ohio Northern since Finkel’s days, the University continues to attract brilliant problem-solvers molded in Finkel’s character who graduate to become leading mathematicians and physicists. 

Read about three outstanding alumni from ONU’s School of Science, Technology, and Mathematics, housed in the Getty College of Arts & Sciences, who are making a mark in the world of physics and mathematics.

From algebra to plasma physics

Thomas (Tommy) Steinberger, BS ’14, Ph.D., is breaking new ground in experimental plasma physics at West Virginia University in Morgantown, W.V.

A research assistant professor in the Department of Physics and Astronomy, he explores charged gas systems (plasmas), investigating gas particle motion and temperature. His work aims to enhance electronics manufacturing, space travel, and our understanding of space phenomena, such as magnetic reconnection.

“I come into the lab and can work on 10 different projects before lunch,” he said. “I have a lot of ongoing projects that are unique. Most of my effort is helping all these projects move forward.”

Steinberger’s journey began in freshman Algebra class in high school. While his classmates bellyached about having to solve 50 of the 100 algebra problems listed in the final exam, he felt a burst of excitement.

 “I completed nearly all 100 in a couple of days, almost obsessively,” he recalled. 

His ONU experience further ignited his passion for problem solving. He joined ONU-SOLVE, a problem-solving group of students that tackle the challenging problems found in leading math magazines such as Fibonacci Quarterly, Mathematics Magazine, The College Mathematics Journal, and Finkel’s own The American Mathematical Monthly.

ONU-SOLVE has been recognized multiple times in recent years for submitting correct solutions, and several have been published in leading journals as the most well-written solutions, according to Mihai Caragiu, Ph.D., professor of mathematics.

According to Steinberger, several solutions he worked on with the assistance of ONU professors received honorable mentions in academic magazines.

“My time at ONU really fostered my interest in math and its application to other fields,” he said. “I received fantastic instruction from professors who truly cared about their students.” 

After graduating from ONU with a double major in mathematics and physics, and minor in astronomy, Steinberger received a master’s degree in physics and astronomy, and a Ph.D. in experimental plasma physics from West Virginia University. 

“The challenge of the subjects (math and physics) resonates with my stubborn nature,” he added. “It leads me to be ever more persistent in my studies.”

From jigsaw mastery to national defense

Ashley Ernst, BS ’15, Ph.D., is a senior physicist at Arcfield in Colorado Springs, Colo, where she helps to solve complex challenges in support of the United States’ most critical national security missions.

Arcfield is a leading provider of mission-focused systems engineering and integration capabilities to the U.S. Intelligence Community, Department of Defense, and other agencies.

Ernst is currently working primarily with simulation and modeling of radiation in both vacuum and in atmosphere or material. She develops technical documentation, presents work to clients, and performs calculations with the aid of software.

“There is always another problem on the horizon,” she says. “The quest for the solution to the next problem is what drives me every day.”

Growing up, Ernst loved to solve jigsaw puzzles with her mom. As her skill increased, so did her hunger for harder challenges.

“They became larger in puzzle number, smaller in puzzle piece size, and more complex in shape and design,” she said. “When that wasn’t enough, I started solving puzzles without guide pictures.”

When she first arrived at ONU, however, she lacked focus and felt extremely homesick. Her ONU professors noticed her struggles and made special efforts to engage her in math conferences and ONU-SOLVE. Professor Caragiu spurred her mind into action by exposing her to Graph Theory and Discrete mathematics.

“Having a direction for my mental energy definitely helped me,” she said.

Within a short time, she says, she was on a better path forward. “I would not be where I am today without their help. The math and physics programs at ONU nurture the search for knowledge. No matter the level, the program meets the student at that level and pushes them to the next level.”

After graduating from ONU with a double major in physics and applied mathematics, Ernst earned a master’s degree in physics and a Ph.D. in hadronic physics from Florida State University. Her second year of graduate school, she received a highly-competitive National Science Foundation (NSF) Graduate Research Fellowship, which she credits to the quality of education and one-on-one mentoring she received at ONU.

“Each student that passes through the math program at ONU is instilled with a sense of excitement regarding a problem. While that problem may change, that excitement stays,” she said. 

From twisty puzzles to cosmic enigmas

Matthew Golden, BS ’17, Ph.D., is a postdoctoral fellow at the Georgia Institute of Technology in the Xtreme Astrophysics group. The group is led by two founding members of the Event Horizon Telescope Collaboration, which released the first image of a black hole in 2019.

“I am a full-time researcher,” he said. “My research focuses on the interface of machine learning and physics. Specifically, I work on using machine learning to accelerate human learning. Our goal is to produce physics equations directly from complex data with minimal human intervention.” His recent publication in Science Advances showed how machine learning learned the equations of a living fluid directly from a video of the experiment.

In high school, Golden became enthralled with solving “twisty puzzles”—think Rubik’s cube, only the more complicated versions. He had puzzles of every platonic solid and with hundreds of pieces. Some would take him mere minutes to solve, others weeks.

“I eventually went on to solve the four-dimensional 3x3x3x3 Rubik’s cube,” he said. “You can find my name in the 4D Hall of Fame as solver #196.”

At ONU, his obsession switched to General Relativity. He spent many late nights in the Mathile Center for the Natural Sciences working through derivations. Then, he’d head to the third floor of Heterick Memorial Library, pull a random math or physics book from the shelf, and read until he was “hopelessly confused.” 

“My schooling was significantly accelerated compared to the usual undergraduate,” he said. “That’s because the physics department was small and the teachers eager to teach at any pace.”

His professors allowed him to take courses in any order he desired. He completed Quantum Mechanics his first semester, then continued to grow his knowledge in leaps and bounds. He’ll never forget being the only student in Dr. Khristo Boyadzhiev’s Real Analysis class. Dr. Boyadzhiev, who obtained YouTube fame for his consistent classroom outfit and greeting, passed away in June 2023.

“It feels like half the people my age know Dr. Khristo Boyadzhiev because of his lovable appearance on Vine,” said Golden. “He was a micro celebrity, and I still remember him laughing about it in the hall.”

Golden also recalls leading a small group of students to first place in the 2017 Ohio MAA Leo Schneider Team Math Competition, dethroning Case Western Reserve University for the first time in many years. 

With accelerated learning and one-on-one attention, Golden says he was way ahead of his peers in graduate school. 

“The education I received was passionate and tailored to me,” he said. “There is no math program in the Midwest that could compete with the personal attention I received at ONU, and it paid off.”

He’s ecstatic that his professional career is centered on gravity research.

“I love that every day I get to think about machine learning and extreme astrophysical environments,” he said. “I get to interact with some of the greatest minds in physics.”

"Model Theory (Ax's Theorem)" by Piotr Kowalski

 

A Fall 2023 full messy chalkboard - random graphs

 

Quanta Magazine: 2023's Biggest Breakthroughs in Physics

 

Quanta Magazine: 2023's Biggest Breakthroughs in Math

Florian Richter: Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions

Topic: Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions 

Speaker: Florian Richter, Northwestern University

June 4, 2020

Ramsey Numbers asymptotics - R(4,t)

An amazing new result

The asymptotics of r(4,t) - by Sam Mattheus and Jacques Verstraete

Random Graphs

Math circle session on random graphs led by Alon Amit at the 2012 Summer BACT

Cellular Automata: Rule 110 + Conway’s Game of Life

"A 1D cellular automaton, Rule 110 (bottom), being fed as input to a 2D cellular automaton, Conway’s Game of Life (top)"

California Approves Revised Math Framework as a Step Forward for Equity and Excellence - July 12, 2023

https://www.cde.ca.gov/nr/ne/yr23/yr23rel54.asp

CDE/SBE Joint Release: #23-54
July 12, 2023

Contact: Communications
E-mail: communications@cde.ca.gov
Phone: 916-319-0818

California Approves Revised Math Framework as a Step Forward for Equity and Excellence

SACRAMENTO—The California State Board of Education today approved the 2023 Mathematics Framework for California Public Schools, instructional guidance for educators that affirms California’s commitment to ensuring equity and excellence in math learning for all students.

“I’m thankful for everyone who worked tirelessly to develop this framework to ensure California’s students have equitable access to rigorous and high-quality math instruction that will prepare them for the future. The framework has struck a great balance in new ways to engage students in developing a love for math while supporting those on an accelerated path,” said Mary Nicely, Chief Deputy Superintendent of Public Instruction. “Our State Superintendent is a champion of equity and excellence, and it is our core mission that every child—regardless of race, ZIP code, or background—has access to a quality education. The approval of the revised Math Framework is one more step forward to meeting the needs of all California’s students.”

The vote today concludes four years of work to update math instructional guidance aligned with the California Common Core State Standards for Mathematics (PDF), which are rigorous learning standards that detail what every student should know and be able to do at every grade level. The framework approved today is the third iteration and reflects revisions responsive to thousands of public comments fielded over two 60-day public comment periods and two public hearings.

The draft was presented by Dr. Mike Torres, Executive Director of the Instructional Quality Commission and a former high school math teacher. Others who participated in the presentation include Dr. Kyndall Brown, Executive Director of the California Mathematics Project at the University of California, Los Angeles; Omowale Moses, Founder and Chief Executive Officer of Math Talks; Dr. Adrian Mims, Founder of The Calculus Project; Ellen Barger, Chair of the Curricular and Improvement Support Committee of the California County Superintendents; and Dr. Linsey Gotanda, Vice Chair of the Instructional Quality Commission.

“This framework provides strategies to challenge, engage, and support all students in deep and relevant math learning by building on successful approaches used in nations that produce high and equitable achievement in math,” said State Board President Linda Darling-Hammond. “It also draws on the experiences of educators who have worked for a decade to develop successful strategies for teaching California’s rigorous standards, carrying those lessons to others across the state. This framework provides teachers and schools with a path to greater excellence with greater equity.”

The guidance includes strategies to:

• Structure the teaching of the state’s math standards around “big ideas” that integrate rather than isolate math concepts—a best practice in high-performing countries.
• Increase focus on developing student mathematical expertise as described in California’s Standards for Mathematical Practice, which include the ability to make sense of problems and persevere to solve them; to reason abstractly and quantitively; to attend to precision; and to apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.
• Connect learning to the “real world” through authentic examples and use of data, prompting students to ask and answer meaningful questions. Adding authenticity to lessons helps teachers answer students’ questions around “why do I need to learn this?”
• Allow students to “see themselves” in curriculum and in math-related careers by making math instruction culturally relevant and empowering.
• Stimulate deep learning by sparking student curiosity through lessons that encourage inquiry and problem-solving.
• Ensure that students develop both appreciation of math concepts and fluency in using math efficiently through the productive use of algorithms and mastery of math facts they have come to understand.
• Integrate and align math concepts taught at the elementary, middle, and high school levels.
• Ensure that all high school math pathways are open to all students.
• Support multiple ways to get more students to higher level mathematics—ranging from successful acceleration to differentiated instruction, personalized supports, extra lab sections, and additional coursework offered at multiple junctures—augmenting more effective core instruction.
• Expand high school math course options to encourage more students to go beyond minimum course requirements.
• Encourage students across age spans to become proficient at understanding and using data—a key skill in the 21st century job market.
• Help students to identify misleading uses of data and use data to make decisions in their roles as global citizens.
• Develop in students a “growth mindset” about mathematics, in line with the groundbreaking research of Stanford’s Dr. Carol Dweck, that supports effort and perseverance.
• Instill confidence in learners by dispelling myths about who can and cannot learn math.
• Develop instruction and curriculum that is “multi-dimensional” and employs the use of visuals, graphics, and words in addition to numbers and equations.

More information is available on the California Department of Education's Mathematics Frameworks web page, which includes frequently asked questions, an overview, and a timeline of events in the framework’s development.

# # # #

Tony Thurmond — State Superintendent of Public Instruction
Communications Division, Room 5602, 916-319-0818, Fax 916-319-0100

Last Reviewed: Wednesday, July 12, 2023

Peter Sarnak on Möbius Disjointness

Senior Capstone presentations - May 10, 2023

Session of Senior Capstone presentations today at ONU, featuring topics on Partial Differential Equations and Number Theory. It was definitely a success, with a lively attendance. Many thanks to these awesome students!

The Patterns in the Primes, with Andrew Granville

 

The Search for Randomness | Jean Bourgain

 

More teaching chalkboards

Chinese Remainder Theorem...

Number theory and cryptography class - chalkboard pictures

Today, the topic was the Euclidean algorithm...

Just before erasing this number theoretic Tibetan mandala :)

Undergraduate research - a post pandemic restart

My first post-pandemic faculty-student paper and the 14-th overall (written with Rachael Harbaugh, ONU '23, a talented Mathematics Education major). Glad for this "restart". Students need confidence, need to be exposed to interesting math topics, and then we hope for the best in their future timelines.

 

Senior Capstone Colloquium - Fall 2022

Mathematics Capstone Colloquium - December 7, 2022 @ohionorthern (from L to R: Dr. Chowdhury, Joelena Brown, Rachael Harbaugh, and Dr. Caragiu)

Joelena Brown: "Rectangular Donut Numbers" (advisor Dr. Chowdhury)

Rachael Harbaugh: "Extending a Putnam Problem to Fields of Various Characteristics" (advisor Dr. Caragiu)

 

Zermelo Fraenkel Choice - with Richard Borcherds

An Invitation to Model Theory - by Jonathan Kirby, UEA

Mobius Randomness and Dynamics - Peter Sarnak

The random Fibonacci sequence and the Viswanath's constant

Random Fibonacci recursion 

https://en.wikipedia.org/wiki/Random_Fibonacci_sequence

In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation ${\displaystyle f_{n}=f_{n-1}\pm f_{n-2}}$, where the signs + or − are chosen at random with equal probability ${\displaystyle {\tfrac {1}{2}}}$, independently for different ${\displaystyle n}$. By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943...(sequence A078416 in the OEIS), a mathematical constant that was later named Viswanath's constant.

Practice test for MATH 3701 - Complex Systems

 

MATH 2651 - two applications of diagonal forms (functional calculus)

Spring 2022 - 8 am class!

 

 

Wolfram Alpha

https://www.wolframalpha.com/ 

- worth trying, it's fun!

Delta... (MATH 2651 classroom stuff)

 

Calculus 1 teaching - limits: Factor-Simplify-Retry

 

Abel Prize 2022: Dennis Sullivan

A nice application of Calculus 2

Via @10kdiver 

Take 2 random numbers X and Y between 0 and 1. What's the probability that the integer nearest to X/Y is even? 

Prove that said probability is (5-Pi)/4.

Experimental math fantasy 1

In my classes I often try to communicate the specific sense of wonder arising from mathematical experimentation, to the benefit of undergraduates. Here "phi" represents the Euler's totient function, in a serendipitous trigonometric (albeit nonlinear) mix.

 

Trigonometric chaos - classroom fun

 

covid-19 era chalkboards (12)

That's from our Math computer lab Mathile 208 (I love that classroom!)

What is chaos? A complex systems scientist explains

Tiny changes, like a butterfly’s wing flapping, can be amplified downstream in a chaotic system. Catherine Falls Commercial/Moment via Getty Images

Mitchell Newberry, University of Michigan

Chaos evokes images of the dinosaurs running wild in Jurassic Park, or my friend’s toddler ravaging the living room.

In a chaotic world, you never know what to expect. Stuff is happening all the time, driven by any kind of random impulse.

But chaos has a deeper meaning in connection to physics and climate science, related to how certain systems – like the weather or the behavior of a toddler – are fundamentally unpredictable.

Scientists define chaos as the amplified effects of tiny changes in the present moment that lead to long-term unpredictability. Picture two almost identical storylines. In one version, two people bump into each other in a train station; but in the other, the train arrives 10 seconds earlier and the meeting never happens. From then on, the two plot lines might be totally different.

Who doesn’t meet in the crowd if the train arrives a few seconds sooner? urbancow/E+ via Getty Images

Usually those little details don’t matter, but sometimes tiny differences have consequences that keep compounding. And that compounding is what leads to chaos.

A shocking series of discoveries in the 1960s and ‘70s showed just how easy it is to create chaos. Nothing could be more predictable than the swinging pendulum of a grandfather clock. But if you separate a pendulum halfway down by adding another axle, the swinging becomes wildly unpredictable.

Chaos is different from random

As a complex systems scientist, I think a lot about what is random.

What’s the difference between a pack of cards and the weather?

You can’t predict your next poker hand – if you could, they’d throw you out of the casino – whereas you can probably guess tomorrow’s weather. But what about the weather two weeks from now? Or a year from now?

Randomness, like cards or dice, is unpredictable because we just don’t have the right information. Chaos is somewhere between random and predictable. A hallmark of chaotic systems is predictability in the short term that breaks down quickly over time, as in river rapids or ecosystems.

Chaos can explain why climate is predictable while weather isn’t. Sören Lubitz Photography/Moment via Getty Images

Why chaos theory matters

Isaac Newton envisioned physics as a set of rules governing a clockwork universe – rules that, once set in motion, would lead to a predetermined outcome. But chaos theory proves that even the strictest rules and nearly perfect information can lead to unpredictable outcomes.

This realization has practical applications for deciding what kinds of things are predictable at all. Chaos is why no weather app can tell you the weather two weeks from now – it’s just impossible to know.

On the other hand, broader predictions can still be possible. We can’t forecast the weather a year from now, but we still know what the weather is like this time of year. That’s how climate can be predictable even when the weather isn’t. Theories of chaos and randomness help scientists sort out which kinds of predictions make sense and which don’t.

Read other short accessible explanations of newsworthy subjects written by academics in their areas of expertise for The Conversation U.S. here.

Mitchell Newberry, Assistant Professor of Complex Systems, University of Michigan

This article is republished from The Conversation under a Creative Commons license. Read the original article.