-Several theorems on sufficient unimodality conditions are formulated for a sum of k normal distributions with the same variance and with different mean values µ i , i = 1, …, k , 2 ≤ k < ∞ , taken with their a priori probabilities π i . On the basis of these theorems, estimates for the lower and upper bounds for the mode numbers m are obtained for k ≥ 3 in the case when the mixture contains k * components, 2 ≤ k * < k , satisfying the unimodality conditions.