Elliptic curves and the number of distinct residues of x2 + bx + c/x modulo p

For any prime p > 3 and rational p -integers b , c with c ( b 3 − 27 c ) ≢ 0 ( mod p ) let V p ( x 2 + b x + c x ) be the number of distinct residues of x 2 + b x + c x modulo p as x runs over 1 , 2 , … , p − 1 . In this paper, we reveal the connection between V p ( x 2 + b x + c x ) and the number of points on a certain elliptic curve over the field F p of p elements.