Cyclic affine planes of even order
Let a be an integer. If t is any integer with \((t,a)=1\), \(\exp_ a(t)\) would mean the smallest positive integer \(\ell\) such that \(t^{\ell}\equiv 1 (mod a)\). Let (p/q) be the Legendre symbol. The main purpose of the paper is to prove the following result. Suppose that there exists a cyclic affine plane of even order n. Then (1) either \(n=2\) or \(n\equiv 0 (mod 4);\) (2) for each prime divisor p of n, we have either \((p/q)=1\) for each prime q, \(q| n^ 2-1\), and for some positive integer r, \(n+1| p^ r+1\) and \(n-1| p^ r-1\), according as \(\exp_{n^ 2-1}(p)\) is odd or even.
scientific article; zbMATH DE number 4116552