Quadratic residues and quartic residues modulo primes.

In this paper, we study some products related to quadratic residues and quartic residues modulo primes. Let p be an odd prime and let A be any integer. We determine completely the product f p (A) : = ∏ 1 ≤ i , j ≤ (p − 1) / 2 p ∤ i 2 − A i j − j 2 (i 2 − A i j − j 2) modulo p ; for example, if p ≡ 1 (mod  4) then f p (A) ≡ − (A 2 + 4) (p − 1) / 4 (mod  p) if A 2 + 4 p = 1 , (− A 2 − 4) (p − 1) / 4 (mod  p) if A 2 + 4 p = − 1 , where (⋅ p) denotes the Legendre symbol. We also determine ∏ i , j = 1 p ∤ 2 i 2 + 5 i j + 2 j 2 (p − 1) / 2 (2 i 2 + 5 i j + 2 j 2 ) and ∏ i , j = 1 p ∤ 2 i 2 − 5 i j + 2 j 2 (p − 1) / 2 (2 i 2 − 5 i j + 2 j 2 ) modulo p. [ABSTRACT FROM AUTHOR]