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Based on the fundamental unit of      Q ( n ) Q\left ( {\sqrt n } \right )      , an arbitrary real quadratic field, we provide a necessary condition for the class number      h ( n ) h\left ( n \right )       to be 1. For      n = 4 m 2 + 1 n = 4{m^2} + 1       we prove the equivalence of three necessary  and  sufficient conditions for      h ( n ) h\left ( n \right )       to be 1. One of these conditions is that      −  x 2 + x + m 2 - {x^2} + x + {m^2}       is prime for all integers      x x       such that      1 &gt; x &gt; m 1 &gt; x &gt; m      . This is the exact analogue of the complex quadratic field case. We discuss the connection with a conjecture of S. Chowla as well as with other related topics.

Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla