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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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DOI

### Online Encyclopedia of Integer Sequences:

Maximal number of edges in a graceful graph on n nodes.The smallest cardinality of a difference-basis in the cyclic group of order n.

Irregular triangle read by rows: representative simple difference sets of Singer type of order m, for m = A000961(n), for n >= 1.

Number of classes of simple difference sets of the Singer type (m^2 + m + 1, m + 1, 1) with m = m(n) = A000961(n), for n >= 1.