The largest prime factor of X3+2
The largest prime factor of X3+2
复制标题
X3 2 的最大素因数
DOI：
10.1112/plms/82.3.554
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发表时间：
2001
影响因子：
1.8
通讯作者：
D. R. Heath
中科院分区：
文献类型：
--
作者：
D. R. Heath
Hooley于1978年研究了X3+2的最大素因子，并给出了一个条件证明，即它无限经常至少与X1+δ一样大，并且具有一定的正常数δ。当 δ=0 时获得这样的结果是微不足道的。人们可能会认为胡利的结果是 X3+2 无限常为素数这一猜想的近似。 Hooley 所要求的条件，即他的 R* 猜想，给出了短 Ramanujan-Kloosterman 和的非平凡界限。本文无条件证明了 X3+2 的最大素因子无限经常至少与 X1+δ 一样大，尽管其常数比 Hooley 获得的常数小得多。为了做到这一点，我们证明了具有平滑模的短 Ramanujan-Kloosterman 和的非平凡界限。还需要修改 Hooley 使用的切比雪夫方法，以确保出现的总和确实具有足够平滑的模数。 2000 年数学学科分类：11N32。
The largest prime factor of X3+2 was investigated in 1978 by Hooley, who gave a conditional proo that it is infinitely often at least as large as X1+δ, with a certain positive constant δ. It is trivial to obtain such a result with δ=0. One may think of Hooley's result as an approximation to the conjecture that X3+2 is infinitely often prime. The condition required by Hooley, his R* conjecture, gives a non‐trivial bound for short Ramanujan–Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of X3+2 is infinitely often at least as large as X1+δ, though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non‐trivial bound for short Ramanujan–Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus. 2000 Mathematics Subject Classification: 11N32.