On generalized quadratic equations in three prime variables |
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Authors: | Man-Cheung Leung Ming-Chit Liu |
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Affiliation: | 1. Department of Mathematics, University of Hong Kong, Pokfulam Rd., Hong Kong
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Abstract: | Leta 1,a 2 anda 3 be any nonzero integers which are relatively prime and not all negative. In this paper, as a parallel problem of [11] for each integerk≥2, we consider the setE(X) of positive integersb≤X which satisfy the condition of congruent solubility and that the equation $$a_1 p_1^2 + a_2 p_2^2 + a_3 p_3^k = b$$ is insoluble in primesp j. We obtain CardE(X)≤X 1-ε. Our result extends the wellknown classical results (by Legendre and Gauss and byDavenport andHeilbronn [2]) on the equation $$x_1^2 + x_2^2 + x_3^k = b$$ in integral variablesx j with the above bound for CardE(X) better than that in [2]. |
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