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Characterization of power digraphs modulo n

Uzma Ahmad, Syed Husnine (2011)

Commentationes Mathematicae Universitatis Carolinae

A power digraph modulo n , denoted by G ( n , k ) , is a directed graph with Z n = { 0 , 1 , , n - 1 } as the set of vertices and E = { ( a , b ) : a k b ( mod n ) } as the edge set, where n and k are any positive integers. In this paper we find necessary and sufficient conditions on n and k such that the digraph G ( n , k ) has at least one isolated fixed point. We also establish necessary and sufficient conditions on n and k such that the digraph G ( n , k ) contains exactly two components. The primality of Fermat number is also discussed.

Congruences for certain binomial sums

Jung-Jo Lee (2013)

Czechoslovak Mathematical Journal

We exploit the properties of Legendre polynomials defined by the contour integral 𝐏 n ( z ) = ( 2 π i ) - 1 ( 1 - 2 t z + t 2 ) - 1 / 2 t - n - 1 d t , where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer r , a prime p 5 and n = r p 2 - 1 , we have k = 0 n / 2 2 k k 0 , 1 or - 1 ( mod p 2 ) , depending on the value of r ( mod 6 ) .

Congruences for q [ p / 8 ] ( m o d p )

Zhi-Hong Sun (2013)

Acta Arithmetica

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine q [ p / 8 ] ( m o d p ) in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.

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