Email: shapiro@math.ohio-state.edu.
For OSU sites try: Department of Mathematics or The Ohio State University
I am involved with the Ross Mathematics Program, a summer math camp for high school students who are deeply interested in mathematics. It is truly an excellent experience for ambitious young people with mathematical interests.
Here are some Short Articles intended for bright middle school students.
DIVISIBILITY PROPERTIES OF INTEGER SEQUENCES.
For a sequence f of nonzero integers and 0 ≤ k ≤ n, define the f-nomial coefficient [n | k] as the fraction:
[n | k] f := ( f(n)f(n-1) . . . f(n-k+1) ) / ( f(k)f(k-1) . . . f(1) ).
Let Δ(f) be the infinite triangle of those numbers [n | k] f . When I is the sequence (1, 2, 3, . . . ) then Δ(I) is Pascal’s Triangle.
Define f to be “binomid” if every entry of Δ(f) is an integer.
Surprisingly, each row and each column of Pascal’s triangle is also binomid. Those numerical triangles fit together to provide the binomid pyramid BP(f).
Definition. Sequence f is “binomid at every level” if every row and column of Δ(f) is binomid. Equivalently: every entry of the pyramid BP(f) is an integer.
For example, define I(n) = n, G(n) = 2^n – 1, and F(n) = n-th Fibonacci number. Then sequences I, G, and F are binomid at every level.
For generalizations and proofs see: “Divisibility properties for integer sequences,” Integers 23 (2023), #A57.
Earlier versions: Divisibility Properties and https://arxiv.org/abs/2302.02243 .
TOWERS OF POWERS.
We consider sequences in Z/kZ, the system of integers modulo k.
For instance, the sequence 11, 22, 33, . . . , nn is eventually periodic (mod k). What is its minimal period?
Which integers c can be expressed as c = xx (mod k) ?
Do similar properties hold for the sequence 111, 222, 333, . . . in Z/kZ ?
For fixed n what about the sequence n, nn, nnn, . . . , in Z/kZ ?
Details appear in the paper: Iterated Exponents, published in the journal Integers: Electronic Journal Of Combinatorial Number Theory 7 (2007) #A23.
SUMS OF SQUARES IDENTITIES.
Suppose K is a field and let DK(n) be the set of nonzero elements of K that are expressible as a sum of n squares in K. Certainly DK(1) is closed under multiplication since it’s just the set of all nonzero squares. The set DK(2) is also closed under multiplication. That closure is clear from the following 2-square identity:
(x12 + x22)(y12 + y22) = (z12 + z22),
where z1 = x1y1 + x2y2 and z2 = x1y2 – x2y1.
Questions. For which n is DK(n) multiplicatively closed, for every field K?
When is there an n-square identity where each zk is a bilinear expression in the sets of variables X and Y ?
Euler recorded a 4-square identity, which is related to the later discovery (invention?) of quaternions by Hamilton in 1843. Soon afterwards Graves and Cayley found the octonions, an 8-dimensional (non-associative) algebra whose norm provides an 8-square identity. In 1898 Hurwitz used linear algebra to answer our bilinear question, proving his
“1, 2, 4, 8 Theorem”. Further details on the history of this problem and its generalizations appear in the following lecture notes on “Products of Sums of Squares”. Those expository lectures were part of a mini-course given at the Universidad de Talca (Chile) in December 1999.
Lecture 1: Introduction and History.
Lecture 2: Integer Compositions.
Lecture 3: Formulas over Arbitrary Fields.
Those notes provide an introduction to the more extensive treatment of this subject given in the book:
D. B. Shapiro, Compositions of Quadratic Forms, W. de Gruyter Verlag, 2000.
This book is available electronically here.