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

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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