Bounding the 3-part can be reduced to the problem of counting the number
of squares of the form , where is a square-free positive
integer, and and are integers in the ranges , . This counting problem is nontrivial because of the disproportionate ranges of the variables. We show that using a variant of the square sieve in combination with the q-analogue of van der Corput's method allows one to tackle such a counting problem successfully, giving a
nontrivial upper bound for the 3-part of class numbers of quadratic
fields. This new method of counting integer points is quite general, and
has recently been used by Heath-Brown to give an upper bound for the size of discriminants of imaginary quadratic fields whose class group can have exponent 5.