Let a, b be non-zero integers. Let p1,…,pr be distinct primes, and let P= {± |m1,…, mr are nonnegative integers}. Further let K be an algebraic number fieldof degree n with n ≥3, and let hk denote the class number of K. For α1,…, αm∈K with1 < m < n, let △(α1,… ,αm) denote the discriminant of α1,…,αm, and f(x1,… , xm) =μNK/Q(α1x1 +… + αmxm) ∈ Z[x1,…, xm]. In this paper we protve that if f(x1,…, xm) is non-degenerate and pi + △(α1,…,αm) for i=1,…, r, then the equation f(x1,…, xm) = by has at most (4sd2)2 (sd)6 (sd)6 integer solutions (x1,…,xm) satisfy gcd(x1,…,xm) = 1 andy∈P, where d= n!, s = r + ω(b) and ω(b) is the number of distinct prime factors of b.
