Lower Bound for Class Numbers of Certain Real Quadratic Fields

Let d be a square-free positive integer and h(d) be the class number of the real quadratic field Q(root d). We give an explicit lower bound for h(n(2)+r), where r= 1,4. Ankeny and Chowla proved that if g > 1 is a natural number and d=n(2g)+1 is a square-free integer, then g |h(d) whenever n > 4. Applying our lower bounds, we show that there does not exist any natural number n > 1 such that h(n(2g)+ 1) =g. We also obtain a similar result for the family Q(root n(2g)+4). As another application, we deduce some criteria fora class group of prime power order to be cyclic