Abstract
Let a 1,..., a 9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a 1,..., a 9 are pairwise coprime and a 1 + · · · + a 9 ≡ b (mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a 1,..., a 9, n) such that the cubic equation a 1 p 31 + · · · + a 9 p 39 = b is solvable with p j ≪ P for all 1 ≤ j ≤ 9. It is proved that one can take P = max {|a 1|, ..., |a 9|}c with c + |b|1/3 with c = 2. This improves upon the earlier result with c = 14 due to Liu (2013).
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Zhao, L. Small prime solutions to cubic equations. Sci. China Math. 59, 1909–1918 (2016). https://doi.org/10.1007/s11425-016-5150-5
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DOI: https://doi.org/10.1007/s11425-016-5150-5