A diophantine equation concerning the divisibility of the class number for some imaginary quadratic fields

AbstractLet d, d1, d2 ϵ N be square free with d=d1d2, and let h(-d) and IK denote the class number and the class group of K=Q(−d) respectively. Let p be an odd prime with p>3. In this paper we prove that if d≥exp exp exp 105, d1a2 + d2b2 = δkp,a, b, k ∈ N, b = 2m, m ≥ 0,gcd(d1a, d2b) = 1, k> 1, δ ∈ {1,4} and d1 + d2 ≢ 0 (mod 8) for δ = 1, then h(-d) ≡ 0 (mod pg) except an explicit case, wh g is the number of different genera of IK