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Yildirim, Small gaps between products of two primes
 20世纪最伟大数学家埃尔德什开始关注这个问题, 但 也没得出有用结果。最近国际顶尖数学 家 Goldston 等正在研究这个问题。得到国际数学界广泛的支持和关注，但文章都发表在著 名杂志上，但没有得出任何实质性进展，蒋春暄在 2002年[1]就彻底证明了它，但国内外数 学家都读了它，都不说话，看到文献[4]后，我们决定写本文，如不用 Jiang函数，再过两百 年也不一定能证明它，国内更无人研究它，这才是研究方向！2009年 1月 10日蒋春暄为休 息去参加宋正海讲座在公共汽车上发现公式(22),1 月 1
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Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
Abstract

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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.