On the Waring–Goldbach Problem for Fourth and Fifth Powers
On the Waring–Goldbach Problem for Fourth and Fifth Powers
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DOI：
10.1112/plms/83.1.1
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发表时间：
2001-07
影响因子：
1.8
通讯作者：
K. Kawada;T. Wooley
中科院分区：
文献类型：
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作者：
K. Kawada;T. Wooley
证明了与14模240同余的每个足够大的整数可以写成14个4次方素数的和，每个足够大的奇数可以写成21个5次方素数的和。各自的隐式边界14和21在前面的边界15(根据Davenport的工作)和23(由于Thanigasalam)的基础上得到了改进。这些结论是通过Hardy-Littlewood方法建立的，证明有些新颖，因为它们使用了直接由素数上的指数和与线性筛子相结合的估计，而不是传统的方法，这种方法通过将较小的圆弧估计转化为基于不限于素数的变量的辅助均值估计来“浪费”一个或两个变量。在关于五次幂的工作中，将转换原理应用于涉及几乎素数的同余问题，以得到仅涉及素数的期望的结论。2000数学科目分类：11P05、11N36、11L15、11P55。
It is shown that every sufficiently large integer congruent to 14 modulo 240 may be written as the sum of 14 fourth powers of prime numbers, and that every sufficiently large odd integer may be written as the sum of 21 fifth powers of prime numbers. The respective implicit bounds 14 and 21 improve on the previous bounds 15 (following from work of Davenport) and 23 (due to Thanigasalam). These conclusions are established through the medium of the Hardy‐Littlewood method, the proofs being somewhat novel in their use of estimates stemming directly from exponential sums over prime numbers in combination with the linear sieve, rather than the conventional methods which ‘waste’ a variable or two by throwing minor arc estimates down to an auxiliary mean value estimate based on variables not restricted to be prime numbers. In the work on fifth powers, a switching principle is applied to a cognate problem involving almost primes in order to obtain the desired conclusion involving prime numbers alone. 2000 Mathematics Subject Classification: 11P05, 11N36, 11L15, 11P55.