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We prove a nontrivial bound of  O (| D | 27/56+ε ) for the 3-part of the class number of a quadratic field ℚ(√ D ) by using a variant of the square sieve and the  q -analogue of van der Corput's method to count the number of squares of the form 4 x  3  −  dz  2  for a square-free positive integer  d  and bounded  x, z .

A bound for the 3-part of class numbers of quadratic fields by means of the square sieve