Abstract. We prove that if x m + ax n permutes the prime field F p , whereprove that if q ≥ 4 and m > n > 0 are fixed and satisfy gcd(m − n, q − 1) > 2q(log log q)/ log q, then there exist permutation binomials over F q of the form x m + ax n if and only if gcd(m, n, q − 1) = 1.
We consider the algebraic curve defined by y m = f (x) where m ≥ 2 and f (x) is a rational function over F q . We extend the concept of pure gap to c-gap and obtain a criterion to decide when an s-tuple is a c-gap at s rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.
Abstract. In this paper we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices Lu = 1 0 u 1 and Rv = 1 v 0 1 , where u and v are nonnegative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numeratordenominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree.
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