Daniel Shapiro's Home Page

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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:

  • (614) 688-1488 office phone
  • Mailing address:
    Department of Mathematics
    231 West 18th Avenue
    The Ohio State University
    Columbus, OH 43210-1174

    SHORT ARTICLES for high school (and middle school) students.

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

    Walking the dog.

    Tiling problems.

    Measuring liquids.

    Torus Trek.

    Big Numbers.

    Drawing Stars.

    Four Numbers Game.

    Nonstandard Digits.


    We consider questions 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 we also investigate properties of the sequence   n,   n^n = nn,   n^(n^n) = nnn, . . . , when reduced (mod k).

    Details appear in the paper:  Iterated Exponents, published in the journal Integers: Electronic Journal Of Combinatorial Number Theory 7 (2007) #A23.


    Suppose K is a field and let DK(n) be the set of nonzero elements of K which are expressible as a sum of n squares in K. Certainly DK(1) is a subgroup of the multiplicative group of K since it is just the set of all nonzero squares. The set DK(2) is also a subgroup because it is 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.
    Question. For which other values of n is the set DK(n) a subgroup for every K?
    That is, when is there an n-square identity?

    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.