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A theorem in finite projective geometry and some applications to number theory. (English) Zbl 0019.00502
Points in a finite projective plane are given homogeneous coordinates in the Galois field $$\mathrm{GF}(p^n)$$. Associating each point with a unique element in $$\mathrm{GF}(p^{3n})$$ it is shown that there exists a collineation $$C$$ such that if $$A$$ and $$B$$ denote points a power of $$C$$ transforms $$A$$ into $$B$$. It follows that points can be arranged in a “regular” rectangular array (an array in which each row contains all points, each column contains all points of a line, each row is a cyclic permutation of the first). Subscripts assigned to the points and so arrayed show that if $$m$$ is a power of a prime there exists a set of $$m+1$$ integers (called a “perfect difference set”) $$d_0, d_1,\ldots, d_n$$ such that the $$m^2+m$$ differences $$d_i-d_j$$ $$(i\neq j;\;i,j = 0, 1, \ldots, m)$$ are congruent, modulo $$q= m^2+m+1$$, to the integers $$1, 2, \ldots, m^2+m$$. This leads to a perfect partition $$a_0, a_1, \ldots, a_m$$ of $$q$$; i. e., each residue class modulo $$q$$ is represented uniquely by a circular sum of the $$a$$’s (sum of consecutive $$a$$’s). The number of perfect difference sets corresponding to a given $$q$$ is discussed. A partial list of perfect difference sets and perfect partitions is given. The concepts are generalized for $$q=p^{kn} + p^{(k-1)n} + \ldots+ p^n + 1$$.
Reviewer: J. L. Dorroh

##### MSC:
 11B75 Other combinatorial number theory 51E20 Combinatorial structures in finite projective spaces 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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