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On generalized quadratic equations in three prime variables

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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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  1. Man-Cheung Leung

    Present address: Department of Mathematics, Columbia University, 10027, New York, NY, USA

Authors and Affiliations

  1. Department of Mathematics, University of Hong Kong, Pokfulam Rd., Hong Kong

    Man-Cheung Leung & Ming-Chit Liu

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  1. Man-Cheung Leung
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  2. Ming-Chit Liu
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Leung, MC., Liu, MC. On generalized quadratic equations in three prime variables. Monatshefte für Mathematik 115, 133–167 (1993). https://doi.org/10.1007/BF01311214

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  • DOI: https://doi.org/10.1007/BF01311214

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