摘要
在加性数论中,人们经常研究将一个正整数表示成素数幂之和的可能性.1937年Vinogradov[1]证明了任何一个充分大的奇数均可表为三个素数的和,这就是著名的三素数定理.对于非线性的情况,1938年,华在[2]中明了:任意充分大的奇整数N≡5(mod24)可以表为5个素数的平方和;任意充分大的奇整数N可以表为9个素数的立方和.对于高次和和混合次方的情形,参看文献[3],[4],[5].
另一方面,研究在适当条件限制条件下的堆垒素数的问题也吸引了大量的数学工作者.在这方面的研究比较多的是将变量限制在小区间中取值,这类问题称为小区间上的堆垒素数问题.比如,在潘([6],[7],[8])的工作基础上,展[9]研究了小区间上的三素数定理,证明了每一个充分大的奇数N都可以表示成其中i=1,2,3.Baker与Harman[10]利用筛法把N的指数从5/8改进到了4/7.关于五个素数平方和定理的小区间问题也有很多结果.首先,人们在广义黎曼猜想的条件下研究这个问题.刘和展[11],每一个充分大的模24余5的正整数N都可以表示成其中U=(?).后来,在1998年,刘和展[12]找到一种新的方法来扩大华林-哥德巴赫问题的主区间.正是这种技巧,研究者可以忽略Siegel零的影响,更好的研究堆垒素数问题.比如:Bauer[13]利用这种技巧在五个素数平方和的小区间问题上得到了U=(?).在2006年,Bauer和王[14]又得到U=(?).不久,吕[16]得到U=(?).最近,刘、吕和展在[17]得到了U=(?).
本文研究这一问题的推广.在[2]中,华证明了每个充分大的正整数N,若N满足如下条件:均可以表示成我们的目的是研究这一问题的小区间情形.当k=1时,吕在[21]中进行了研究;当k=2时,上面已给出了很多结果.我们研究正整数k≥4的情形.
本文的结果如下:
定理设N是充分大的正整数且满足(0.1),K=2~(k-1).对k≥4,U=(?),方程有解.
In the additive theory of prime numbers, one studies the representation of positive integers by powers of primes. In 1937, Vinogradov [1] proved that each sufficiently large integer odd N can be written as the sum of three primes, which is known as the famous three prime theorem. For the nonlinear case, Hua [2] proved that each large integer congruent to 5 modulo 24 can be written as the sum of five squares of primes and each large odd integer can be written as the sum of nine cubes of primes. And for higher or hybrid power case, there are many results referring to [3], [4], [5].
On the other hand, the additive theory of prime numbers with certain conditions appeals to many researchers to study, in which to restrict the variables in short intervals is frequently considered. The problems are called the additive theory of prime numbers in the short intervals. For example, basing on the work of Pan and Pan([7], [8], [26]), Zhan [9] proved that every large odd integer N can be represented aswhere i = 1,2,3. Baker and Harman [10] had reduced the exponent 5/8 to 4/7. There exist many results about the sum of five square primes in the short intervals.
First of all, assuming the Generalized Riemann Hypothesis, Liu and Zhan [11] considered short interval version of this problem. They showed the following result for any sufficiently large integer N satisfying N≡5 (mod 24) can be written aswhere U=(?).
Later on, in 1998, Liu and Zhan [12] found a new approach to treat the enlarged major arcs in the Waring-Glodbach problem, in which the possible existence of the Siegel zero does not have a special influence, and hence the Deuring-Heilbronn phenomenon can be avoided. Due to this approach, they obtained that (1.2) is true for U = (?). This approach has been successfully applied to a number of additive problems concerning primes. Recently Bauer [13] used the approach mentioned above and showed that U = (?). In 2006, Bauer and Wang [14] obtained U =(?). Liu Lu and Zhan in [17] obtain the U = (?) .
In this paper , we study the generalization of the problem. Hua in [2] proved that every sufficiently large integer N satisfying the congruence conditioncan be written asOur goal is to study this problem in short intervals. The case k = 1 was studied by L(?) in [21]. Many results about k = 2 have been worked out by different authors.
We shall prove the following theorem,
Theorem For each sufficiently large integer N as in (0.1), k≥4 and K = 2~(k-1), the equationhas solutions.
引文
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