Small prime solutions to cubic equations
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...
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| Veröffentlicht in: | Science China. Mathematics 2016-10, Vol.59 (10), p.1909-1918 |
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| container_end_page | 1918 |
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| container_issue | 10 |
| container_start_page | 1909 |
| container_title | Science China. Mathematics |
| container_volume | 59 |
| creator | Zhao, LiLu |
| description | 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）. |
| doi_str_mv | 10.1007/s11425-016-5150-5 |
| format | Article |
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| source | Springer Nature - Complete Springer Journals; Alma/SFX Local Collection |
| subjects | Applications of Mathematics Cubic equations Integers Mathematical analysis Mathematics Mathematics and Statistics mod 三次方程 互质 平均 立方型 非零整数 |
| title | Small prime solutions to cubic equations |
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