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Binary quadratic forms with large discriminants and sums of two squareful numbers

Abstract. Let UF ðxÞ be the number of integers not exceeding x that can be represented by a primitive positive definite binary quadratic form F A Z½x; y having discriminant D < 0. It is shown that UF ðxÞ ge jDjÀe xðlog xÞÀ 2 uniformly in jDj e ðlog xÞ log 2Àe and UF ðxÞ ge xðlog xÞÀ1Àkðlogð2kÞÀ1ÞÀe uniformly in jDj e ðlog xÞ 2k log 2Àe for any 1 1 eke À e. 2 1 þ log 2 As an application a problem of Erdos is considered. Let V ðxÞ be the number of in¨ tegers representable as a sum of two squareful integers. Then V ðxÞ g xðlog xÞÀ0:253 . 1. Introduction An integer n is called squareful if pjn implies p 2 jn for all primes p. Since every squareful integer n can uniquely be written as n ¼ a 3 b 2 , m 2 ðaÞ ¼ 1, one easily sees that the 1 zð3=2Þ 1 number of squareful integers not exceeding x satisfies x 2 þ Oðx 3 Þ. Erdos conjectured ¨ zð3Þ that the number V ðxÞ of 1integers not exceeding x that are sums of two squareful integers satisfies V ðxÞ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik (Crelle's Journal) de Gruyter

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