Tom Moore's Web Page
General Information
I am a Professor of Mathematics in the
My degrees were obtained at Stonehill College and at The University of Notre Dame, both schools run by the Congregation of the Holy Cross.
For information on my current courses, office hours, etc. go here.
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Copyright (c) 2000 Scott Kim. All rights reserved.
Mathematical Interests
My research interests are in finite group theory and
number theory, as well as their connections to Recreational
Mathematics. I also have an interest in Cryptography/Cryptology. See my publications
Pedagogical Interests
I am interested in the use of technology at all levels of the
mathematics curriculum.For example in an introductory Linear Algebra course I
have used MINimal MATlab from Joel W.
Robbin which accompanies his 1995 Matrix Algebra textbook. I have
most recently taught this course aided by MAPLE software from Waterloo Maplesoft. I now use
Maple 14 in various of my courses, especially Number Theory.
A great resource for software related to K-12 and collegiate level mathematics
is the Mathematics Archives WWW
Server at the
I am also interested in using recreational mathematics in my
mathematics teaching, whenever possible, especially in Abstract Algebra
courses. There, for example, I illustrate coset partitions using circular
residue designs on the subgroups of the integers mod m or on the group of units
mod m. We analyze peg-jumping games with the aid of the Klein four-group and
sliding marker games on star graphs with the aid of Lagrange's Theorem. In this same
spirit I have recently created a course and a manuscript called Mathematical
Games and Puzzles for K-12 Classrooms (MATC 560).
My graduate student, Neil Roza, is in the process of creating Java Applets for some of the games from my manuscript. At this time you can play Ducci's Game which is a subtraction game. It consists of rounds of subtraction (which the applet does for you) starting with a quadruplet of integers of your choice and with the goal being to arrive at four zeroes in the greatest number of steps. Go here to play, answering "open" to the dialog box you will see displayed.
Also available is the Pile Splitting Game. A natural number N is used as input. Triple click on the starting default value of 1; type your starting value of N; hit enter. The program generates the first split automatically into two summands. To change a summand just triple click on it and then type the value of your choice and hit enter. You then continue to split summands by double clicking on their folder symbols (and you can change the program's generated summands at any time with a triple click and by entering your choice). Continue splitting until you reach all 1s. For each split the applet will multiply the summands and finally it will total all these products. A surprise awaits you as you play again with the same number N and split in different ways! Go here to play, answering "open" to the request in the dialog box you'll see.
Pretty pictures and more! Make your own Residue Designs. This applet illustrates the arithmetic/algebra in the group of reduced residues modulo N. You first specify a natural number N as modulus and a multiplier which must be less than N and relatively prime to N. Do this by clicking on the change numbers button. (The slow draw button option is currently not implemented.) The applet places all the reduced residues equally spaced around a circle. Then it multiplies each residue by your choice of multiplier. If the answer to multiplying x on the circle is y (reduced mod N) on the circle, then a chord is drawn joining x and y. Images can be very intriguing. Go here to create a design, answering "open" to the request in the dialog box you'll see.
The game of SIM. Invented by Gustavus Simmons this game is from graph theory, specifically, the easiest case of Ramsey theory. But it can be played by preschoolers as well as adults! Two players take turns coloring the edges of the complete graph K6, each having a different color to use. The image of K6 we use is that of a hexagon and all its diagonals. The first to draw a triangle (with its corners on three of the hexagon's corners) is the loser. So it's a game of avoidance! Go here to play, answering "open" to the request in the dialog box you'll see. Thanks again to Neil Roza.
Mathematics Association of America
The Math and CS Department here at BSU hosted the annual
November meeting of the Mathematical Association of America-Northeastern
Section on Friday and Saturday, Nov. 16 and 17, 2001, and the theme was...Recreational
Mathematics! See what this national organization devoted to collegiate
level mathematics has to offer: MAA site.
Hear ye! Hear ye!
The Math and CS Department here at BSU will host the annual meeting of the Mathematical Association of America-Northeastern
Section again, this coming November 16-17, 2012, and the theme will be Problems, Problems, Problems: the creation, solution and publication of mathematics problems.
Bill Ritchie, founder of Binary Arts (now Thinkfun), and me. November, 2001 Binary Arts/Thinkfun.
In
January, 2004, I traveled to the Joint Mathematics Meeting in Phoenix where my
colleague Tom Koshy from Framingham State College and I celebrated the
publication of his latest textbook. Here we are pointing out a small
contribution I made to this outstanding discrete mathematics text.
In January,
2006, Tom Koshy and I, and my daughter Katie, traveled to the Joint Mathematics Meetings at
San Antonio, TX, where Tom and I served as facilitators at a session on
Applications of Number Theory. The three of us also traveled in January, 2008 to the Joint Mathematics Meetings in San Diego. In January, 2009, Tom and I were co-facilitators at a session on
The Power and Beauty of Number Theory again at the Joint Mathematics Meetings, this time in Washington, D.C. Speakers included Ken Ono, Carl Pomerance, Kirstin Eisentrager and George Andrews
.
In January, 2006, Tom Koshy and I, and my daughter Katie, traveled to the Joint Mathematics Meetings at San Antonio, TX, where Tom and I served as facilitators at a session on Applications of Number Theory. The three of us also traveled in January, 2008 to the Joint Mathematics Meetings in San Diego. In January, 2009, Tom and I were co-facilitators at a session on The Power and Beauty of Number Theory again at the Joint Mathematics Meetings, this time in Washington, D.C. Speakers included Ken Ono, Carl Pomerance, Kirstin Eisentrager and George Andrews
.
Awards
The Dr. V. James DiNardo Award for Excellence in Teaching .
Bridgewater State College, 1991.
Award for Distinguished College or University Teaching.
Mathematical Association of America, Northeastern Section, 1993.
Here is the full list of awardees for the Northeastern Section of the MAA, which comprises the New England states as well as the four Canadian Provinces of New Brunswick, Newfoundland, Nova Scotia, and Prince Edward Island: 1992
Frank Morgan
Williams College
1993
Thomas Moore
Bridgewater State College
1994
Robert Devaney
Boston University
1995
Thomas Banchoff
Brown University
1996
Colin Adams
Williams College
1997
James J. Tattersall
Providence College
1998
Robert Case
Northeastern University
1999
Charles Vinsonhaler
University of Connecticut
2000
Edward Burger
Williams College
2001
Paul Blanchard
Boston University
2002
Laura Kelleher
Massachusetts Maritime Academy
2003
Emma Previato
Boston University
2004
P. Joseph McKenna
University of Connecticut
2005
David Abrahamson
Rhode Island College
2006
Gilbert Strang
Massachusetts Institute of Technology
2007
Kenneth Gross
University of Vermont
2008
David Carhart
Bentley College
2009
Solomon Friedberg
Boston College
2010
Susan Loepp
WIlliams College
2011
Joseph Silverman
Brown University
.
Publications
A
note on the distribution of primes in arithmetic progressions.
The Journal of Recreational Mathematics, 1972, 5(4), p. 253.
Cayley's Color Group.
The Mathematics Teacher, 1973, 66(7), pp. 615-618.
Order as a subgroup-lattice homomorphism (with H. D'Alarcao).
American Mathematical Society Notices, 1973, 20(2), A-253.
Lagrange's Theorem Revisited (with H. D'Alarcao).
American Mathematical Monthly, 1975, 2(3), pp. 270-273.
Complete residue systems in the sequence of primes (with R. F. Sutherland).
Journal of Undergraduate Mathematics, 1975, 7(1), pp. 49-50.
Euler's formula and a game of Conway's (with H. D'Alarcao).
Journal of Recreational Mathematics, 1976, 9(4), pp. 249-251.
Isomorphisms between groups of rational numbers.
Mathematical Gazette: the Journal of the British Mathematical Association, 1980, 64, pp. 286-287.
An aspect of group theory in residue designs .
Two-Year College Mathematics Readings, Warren Page, editor, The Mathematical Association of America, 1981, pp. 254-257.
What hath Rubik wrought?.
Bridgewater Review, 1984, 2(2), pp. 25-26.
SIM on a microcomputer.
Journal of Recreational Mathematics, 1987, 19(1), pp. 25-29.
Counting bit strings with a single occurence of 00.
Pi Mu Epsilon Journal, 1988, 8(9), pp. 572-575.
Euclid's algorithm and Lame's theorem on a microcomputer.
The Fibonacci Quarterly, 1989, 27(4), pp. 290-295.
On the least absolute remainder Euclidean algorithm.
The Fibonacci Quarterly, 1992, 30(2), pp. 161-165.
Triangle Trek.
Bridgewater Magazine, 1992, 2(4), pp. 12-14.
Was Gauss Smart?
Math Horizons, November, 1999, p. 24.
This article has appeared in a book collection
The Edge of the Universe:
Celebrating Ten Years of Math Horizons
Deanna Haunsperger and Stephen Kennedy, Editors, pub. by the Mathematical Association of America, 2006.
An
Error Found in Lucacs and Tarjan's Mathematical Games (with J. Hatch and M. LaPlante.)
The Journal of Recreational Mathematics, 2003, 31(1),pp.25-28.
On representations of squares using triangular numbers, submitted to
Math Horizons, November, 2011
When two triangular numbers sum to a third , Mathematical Spectrum Vol. 44, Number 2, pg. 50
Mathematical Spectrum, December 22, 2011
Original Problems Published in Mathematics Journals
(1) Prove there are only four Mersenne numbers that are also triangular numbers
The Journal of Recreational Mathematics, 1980, 13(3), p. 218
(2) Prove that the congruence 32k^2+21k+14 = 0 (mod 2^n) always has a solution, for all n >0.
Mathematics Magazine, 1981, 54(3), p. 142.
(3) Show that the equation (2^x-1)(2^y-1)=2^(2^z)+1 is impossible in positive integers x, y and z.
Two Year College Mathematics Journal, 1982, 13(3), p. 208.
(4) If n is an even perfect number then n-phi(n) is the square of an integer, where phi is Euler's phi-function.
Find infinitely many n such that n-phi(n) is a square.
Crux Mathematicorum, 1988, 14(3), p. 93.
(5) Under what conditions on the positive integers a and b will the sides of a nondegenerate triangle be formed
by (i) a, b and gcd(a, b) and (ii) a, b and lcm(a, b)?
Pi Mu Epsilon Journal, 1990, 9(3), p. 199.<
(6) For a < b < c positive integers, if gcd(a, b) =1 and a^2+b^2 = c^2, then we say (a, b, c) is a primitive Pythagorean triple (PPT). If both a and c are primes we call it a prime PPT. (i) If (a, b, c) is a prime PPT, deduce that b = c-1. (ii) Find all prime PPTs in which a and c are twin primes, or both are Mersenne primes, or both are Fermat primes, or one is Mersenne and the other Fermat.
Pi Mu Epsilon Journal, 1993, 9(9), p. 618
(7) Problem 926
Pi Mu Epsilon Journal, 1997, 10(7), pp. 579-580.
(8) Let D(n) be the sum of the (base 10) digits of the positive integer n. Are there twin primes p and p+2 such that D(p) = D(p+2)?
Pi Mu Epsilon Journal, 2001, 11(5), p. 273.
(9) Let t(n) be the nth triangular number, defined by t(n) = t(n-1) + n, for n > 1, with t(1) = 1.
Prove: gcd(t(n-1),t(n))*gcd(t(n),t(n+1)) = t(n), for all n > 1.
Pi Mu Epsilon Journal, 2004, #1081, p. 216.
(10) The number 99
Pi Mu Epsilon Journal, 2008, 12(8), p. 494
(11) Every even perfect number is both the sum and the difference of two distinct deficient numbers.
Pi Mu Epsilon Journal, Fall, 2008, p.560
(12) In what bases are 121, 232 and 343 perfect squares? ,to appear in Crux Mathematicorum, Spring 2011
(13) Integers not the product of two deficients , School Science and Mathematics Journal, Problem 5165, May, 2011
(14) Sophie Germain and Mersenne primes , Math Horizons, a publication of the MAA, Problem 293, Sep, 2011, p. 30
(15) Cubes and even perfect numbers , accepted by and to appear in School Science and Mathematics Journal
(16) Even perfect numbers as fourth powers , accepted by and to appear in School Science and Mathematics Journal
(17) The primality of 9^n+2 and 9^n-2 , accepted by and to appear in Pi Mu Epsilon Journal for (Fall/2012)
(18) Certain simultaneously composite rotation numbers , accepted by and to appear in The Pentagon
(19) Certain exponential trinomials as perfect squares , accepted by and to appear in The Pentagon
(20) Triangular numbers as sums of consecutive squares , accepted by and to appear in Mathematical Spectrum
(21) On a certain diophantine equation , accepted by and to appear in Pi Mu Epsilon Journal
(22) On the compositeness of an exponential expression , accepted by and to appear in Pi Mu Epsilon Journal
(23) Fourth powers and triangular numbers , accepted by and to appear in School Science and Mathematics Journal
(24) Show that a certain automatically generated sequence consists solely of triangular numbers , accepted by and to appear in The Pentagon
(25) On the number of idempotents in a ring with unity , accepted by and to appear in Mathematics Magazine 2012 as QA257
(26) Expressing triangular numbers using a certain number of pentagonal numbers , accepted by and to appear in The Pentagon
(27) On pentagonal numbers not the sum of a certain number of squares , accepted by and to appear in Pi Mu Epsilon Journal
(28) On cubes the sum of two triangular numbers , submitted to Parabola, a publication of the University of New South Wales, AU, November, 2011
(29) Triangular numbers as sums of consecutive integers , submitted to Math Horizons, a publication of the MAA, November, 2011
(30) On cubes that are the sum of Mersenne numbers , accepted and to appear in Mathematical Spectrum
(31) Find infinitely many squares of triangular numbers each the sum of a square and a cube , accepted by and to appear in The Pentagon
(32) On sums of two squares and two cubes , submitted Dec. 6, 2011 to The Pentagon
(33) On squares expressible in a certain binomial form , accepted by and to appear in Pi Mu Epsilon Journal
(34) On squares expressible in a certain trinomial form , accepted by and to appear in School Science and Mathematics Journal
(35) On an algorithmic description of sums of three squares among the integer squares , accepted by The Pentagon (12/15/2011)
(36) On pentagonal numbers and the differences of squares , Math Horizons, problem 274, April, 2012, p. 30.
(37) Bases with at least two repdigits decimally triangular , accepted (12/22/2011) by and to appear in The Pentagon
(38) A base in which certain repdigits are triangular numbers , accepted by and to appear in School Science and Mathematics Journal (12/23/2011)
(39) Fibonacci numbers as differences of hexagonals , submitted to Mathematics Magazine (1/13/2012)
(40) Pentagonal numbers as the difference of hexagonals and vice versa , submitted to The Pentagon (1/14/2012)
(41) On the gcds of triangular and pentagonal numbers of the same rank, accepted by (1/23/2012) and to appear in School Science and Mathematics Journal
(42) An observation on the gcds of powers of 6 and hexagonal numbers , accepted (2/4/2012) by and to appear in the Pi Mu Epsilon Journal
(43) On which Fibonacci numbers are the values of gcds of certain polygonal numbers, accepted (2/2/2012) by and to appear in School Science and Mathematics Journal
(44) Powers of 2 and the differences of pentagonal numbers , submitted (2/12/2012) to Crux Mathematicorum
(45) Two questions on sums of three cubes , submitted (2/14/2012) to Math Horizons
(46) A certain representation of the squares of Fibonacci numbers, submitted (2/17/2012) to The Fibonacci Quarterly
(47) A congruence inspired by Fermat's little theorem , accepted (3/13/2012) by and to appear in School Science and Mathematics Journal
(48) A congruence involving the Euler phi-function , accepted (4/11/12) by and to appear in The Pentagon
Department
of Mathematics and Computer Science
Hart Hall
Bridgewater State University
Bridgewater,MA 02325
U.S.A.
Telephone:
(508) 531-2328
Fax: (508) 697-1361
e-mail: moore@bridgew.edu