Cyclic Affine Planes of Even Order
Document Type
Article
Publication Date
1989
Abstract
In this paper, we prove the following theorem: Suppose there exists a cyclic affine plane of even order n. Then (a) either n=2 or n≡0 (mod4), and (b) for each prime divisor p of n, we have either (pq)=1 for each prime q|n2−1or for some positive integer r (which depends on p), n+1|pr+1 and n−1|pr−1, according as expn2−1(p) is odd or even. For p=2, the former condition cannot hold and hence the latter one holds making expn+1(2) even. As a corollary, we prove that if there exists a cyclic affine plane of order n≡4 (mod8), then (i) n must be a square, (ii) n≡1 (mod3) and (iii) each prime divisor of n+1 is ≡1 (mod4). (For an integer a, if t is any integer with(t,a)=1, expa(t) would mean the smallest positive integer l such that tl≡1 (moda). Here (pq) is the Legendre symbol.
Repository Citation
Arasu, K. T.
(1989). Cyclic Affine Planes of Even Order. Discrete Mathematics, 76 (3), 177-181.
https://corescholar.libraries.wright.edu/math/594
DOI
10.1016/0012-365X(89)90317-8
