On the square-free sieve

Integral points on elliptic curves and 3-torsion in class groups

“…One way to bound the cardinality of E(K, S) is to exploit the fact that, in a certain sense, the points of E(K, S) tend to be separated from each other. This idea is already present in [Sil6], [GS]; let us consider it in the manner of [He,§4]. After some modest slicing of E(K, S), we see that any two points on the same slice are separated by almost 60…”

“…Indeed, for the applications of Section 4, it is not difficult to avoid local heights completely: since we deal with the curves y 2 = x 3 +D, one may use the fact that they are all twists of y 2 = x 3 +1 to prove the required special cases of Proposition 3.4 and Theorem 3.8 in an elementary fashion (cf. [He,Lemma 4.16].…”

The Parity Problem for Reducible Cubic Forms

“…Note, finally, that every time we apply the inequality (2.9), the modulus m is not divisible by any exceptional moduli q > [Z 2 : L : L] < (log N ) α (as otherwise the result to be proven is trivial), we have that m is not divisible by any exceptional moduli q > C(log N ) 2α , where C is an absolute constant, and thus (2.9) is valid. and the ratio of this expression to (y|x)λ(xy(x − y)) can be handled by means of a square-free sieve ( [9], Prop. 3.12).…”

Power-free values, large deviations, and integer points on irrational curves

“…and O f (N (log N ) −7/9 ) are smaller than those in [22], Thm. 5.1 (respectively, O f (N (log N ) −0.8061... ) and O f (N (log N ) −0.6829... )), which were, in turn, an improvement over the bound in [26], Ch.…”