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These mathematical articles are listed below (as pdf files). Your comments are welcome.
Walking the dog.
Four Numbers Game.
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).
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.