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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  &gt; 2 if there exists positive integer  l  such that  d  | ( p l   + 1), where  p  is the characteristic of  F   q   .

On the number of solutions of the equation ( x 1 + . . . + x n ) m = ax 1 . . . x n in a finite field