Let Q be the rational number field and h(m) be the class number of the real quadratic field
with a positive square-free integer m. It is known that if h(m) = 1 holds, then m is one of the following four types with prime numbers p ≡ 1, pt ≡ 3 (mod 4) (1 昤 i ≥ 4) : i) m = p, ii) m = p1, iii) m = 2 or m = 2p2, iv) m = p3p4 (see Behrbohm and Rédei [1]). The sufficient conditions for h(m) > 1 with these m were given by several authors: in all cases by Hasse [2], in case i) by Ankeny, Chowla and Hasse [3] and by Lang [4], in case ii) by Takeuchi [5] and by Yokoi [6].