QUARTIC RESIDUES AND SUMS INVOLVING ( 4 k 2 k )

Let Z be the set of integers, and for a prime p let Zp denote the set of those rational numbers whose denominator is not divisible by p. Let (p ) be the Legendre symbol. Suppose that p is an odd prime and a ∈ Zp. In [7] the author investigated congruences for ∑[p/4] k=0 (4k 2k ) ak modulo p, where [x] is the greatest integer not exceeding x. For k ∈ {0, 1, . . . , p − 1} it is easily seen that p (4k 2k) if and only if 0 ≤ k < p4 or p 2 < k < 3p 4 . In this paper we reveal the connection between quartic residues and the sum ∑[p/4] k=0 (4k 2k ) ak. We also investigate congruences for ∑[3p/4] k=(p+1)/2 (4k 2k ) ak modulo p. Let i = √−1. For an odd prime p let (a+bi p )4 be the quartic Jacobi symbol defined in [1,2,3,4,6]. Following [4] we define