On the number of solutions of the equation (x 1 + . . . + xn ) m = ax 1 . . . xn in a finite field
Yu. N. Baulina · Discrete Mathematics and Applications · 2004
We consider the equation ( x 1 + . . . + x n ) m = ax 1 . . . x n , where a is a nonzero element of the finite field F q , n ≥ 2, and m is a positive integer. Explicit formulas for the number of solutions of this equation in under the condition d ∈ {1, 2, 3, 6}, where d = gcd( m – n, q – 1), are found. Moreover, we obtain formulas for the number of solutions for arbitrary d > 2 if there exists positive integer l such that d | ( p l + 1), where p is the characteristic of F q .