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Research on density estimates for exceptional sets associated with additive representations of natural numbers

与自然数的加性表示相关的异常集的密度估计研究

基本信息

批准号：
17540004
负责人：
KAWADA Koichi
金额：
$ 2.45万
依托单位：
Iwate University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540004/
关键词：
Waring''s Problem Goldbach''s problem Powers of Integers Cubes Biquadrates Prime numbers Almost Primes Waring問題漸近式 Goldbach問題滑らかな数

项目摘要

The most central accomplishment of this three year research project is that we succeeded in determining completely the integers that cannot be written as the sum of at most sixteen biquadrates. We in particular contributed to the proof that every integer exceeding 216th power of 10 and not divisible by 16 can be written as the sum of sixteen biquadrates, by making use of the circle method. We also determined explicitly the exceptional sets associated with the sum of seventeen and eighteen biquadrates. For example, we established that every natural number can be written as the sum of eighteen biquadrates except only for the following seven numbers; 79, 159, 239, 319, 399, 479 and 559. As a by-product, moreover, we provided a new proof, substantially simpler than the previous one, of the fact that every natural number can be written as the sum of at most nineteen biquadrates.Next, concerning the Waring-Goldbach problem, we showed that every sufficiently large integer can be written as the sum of seven cubes of natural numbers that have at most four prime factors counted according to multiplicity. This conclusion improves the previously best result in this direction, in which the upper limit for the number of prime factors was 69 in place of four. The latter theorem was proved by applying sieve methods and the circle method.Besides these two main products mentioned above, we worked on the exceptional sets associated with the following additive problems; sums of seven cubes of smooth numbers (that means natural numbers having smaller primes factors only), sums of fifteen biquadrates, and sums of five cubes of primes. We intend to continue our research in this area.

这个为期三年的研究项目最重要的成就是我们成功地完全确定了不能写成最多16个双方的和的整数。我们特别利用圆法证明了任何大于10的216次方且不能被16整除的整数都可以写成16个双方的和。我们还明确地确定了与17和18双方形和相关的例外集。例如，我们确定，除了以下7个数字外，任何自然数都可以写成18个双方数的和；79、159、239、319、399、479和559。此外，作为一个副产品，我们还提供了一个新的证明，它比前一个证明简单得多，证明了每个自然数最多可以写成19个双方的和。接下来，关于沃林-哥德巴赫问题，我们证明了每一个足够大的整数都可以写成七个自然数的立方数的和，这些自然数根据多重性最多有四个素数因子。这个结论改进了之前在这个方向上的最佳结果，在这个方向上，质因数个数的上限是69，而不是4。用筛法和圆法证明了后一个定理。除了上面提到的两个主要积外，我们还研究了与以下加性问题相关的例外集；平滑数的7个立方数的和（这意味着只有较小质数因子的自然数），15个双平方数的和，以及5个质数立方数的和。我们打算继续在这方面的研究。