摘要
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_1p_1~3+ + a9p_9~3= 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+ |b|^(1/3) with c = 2. This improves upon the earlier result with c = 14 due to Liu(2013).
Let a1,…,a9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a1,…,a9 are pairwise coprime and a1＋…a9≡b （mod 2）. We apply the p-adic method of Davenport to find an explicit P = P（a1,..., a9, n） such that the cubic equation a1p1^3＋…＋a9p9^3=b is solvable with pj≤Pfor all 1≤j≤9. It is proved that one can take P=max（｜a1]…｜a9｜}^c ＋ ｜b｜^1/3 with c=2. This improves upon the earlier result with c=14 due to Liu （2013）.
基金
supported by National Natural Science Foundation of China (Grant No. 11401154)
