An improvement on a theorem of the Goldbach-Waring type.
In this paper the author shows, using the circle method, that all but \(O (x^{19193/19200 +\varepsilon})\) positive even integers up to \(x\) are representable in the form \(p^2_1 + p^3_2 + p^4_3 + p^5_4\) where the \(p_i\)'s are prime numbers. This result sharpens an earlier result of the author in which the exceptional set was of the form \(O(x^{1-\delta})\) with an unspecified positive (small) \(\delta\). The basic structure of the proof here is largely the one used by many authors in this area, which originates from the paper [\textit{M. C. Liu} and \textit{K. M. Tsang}, Théorie des Nombres, C. R. Conf. Int., Québec/Can. 1987, 595--624 (1989; Zbl 0682.10043)]. The exponent \(19193/19200\) obtained here is a consequence of using in a main step, the inequality \([r_2, r_3, r_4, r_5] \geq (r_2 r_3)^{1/7} (r_4 r_5)^{5/14}\) where \([\cdots]\) denotes the least common multiple. Recently a more efficient iterative procedure has been developed which essentially yields \([r_2 , r_3 , r_4, r_5] \approx r_2 r_3 r_4 r_5\). Thus the exponent \(19193/19200\) has now been reduced to \(65/66\) by X. Ren and the reviewer.
scientific article; zbMATH DE number 2012008
scientific article