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 4x3 − dz2 for a square-free positive
integer d and bounded x, z.
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