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The **Ross Mathematics Program** is a six
week residential summer program for high school students talented
in mathematics. It is truly an excellent experience for ambitious young people with
mathematical interests.

Here's a link to the
Ross Program Web Page.

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Further information for D. Shapiro:

Department of Mathematics

231 West 18th Avenue

The Ohio State University

Columbus, OH 43210-1174

USA

These mathematical articles are listed below (as *pdf* files).
Your comments are welcome.

We consider questions in **Z**/k**Z**, the system of integers modulo k.

For instance, the sequence 1^{1}, 2^{2}, 3^{3}, . . . , n^{n}
is eventually periodic (mod k). What is its minimal period?

Which integers c can be expressed as c = x^{x} (mod k) ?

Do similar properties hold for the sequence 1^{11},
2^{22}, 3^{33}, . . . in **Z**/k**Z** ?

For fixed n we also investigate properties of the sequence n, n^n = n^{n},
n^(n^n) = n^{nn}, . . . , when reduced (mod k).

Suppose K is a field and let D_{K}(n) be the set of
nonzero elements of K which are expressible as a sum of n squares in
K. Certainly D_{K}(1) is a subgroup of the multiplicative group of K
since it is just the set of all nonzero squares. The set D_{K}(2) is
also a subgroup because it is closed under multiplication. That
closure is clear from the following **2-square identity**:

(xwhere z_{1}^{2}+ x_{2}^{2})(y_{1}^{2}+ y_{2}^{2}) = (z_{1}^{2}+ z_{2}^{2}),

That is, when is there an n-square identity?Question.For which other values of n is the set D_{K}(n) a subgroup for every K?

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 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 (dvi), (pdf) : Introduction and History.

Lecture 2 (dvi), (pdf) : Integer Compositions.

Lecture 3 (dvi) , (pdf) : 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.