The primes contain arbitrarily long polynomial progressions
The primes contain arbitrarily long polynomial progressions
复制标题
DOI：
10.1007/s11511-008-0032-5
复制
发表时间：
2006-10
期刊：
影响因子：
3.7
通讯作者：
T. Tao;T. Ziegler
中科院分区：
文献类型：
--
作者：
T. Tao;T. Ziegler
我们建立了素数中无穷多项式级数的存在性；更准确地说，给定任何整值多项式P1，…，pk∈Z[m]与P1(0) = …在一个未知数中 =pk(0) = 0，并且给定任意ε&> 0，我们证明了存在无穷多个整数xandm，其中，使得x+P_1(M)，…，x+Pk(M)同时是素数。这些论点是基于[18]中处理线性情形Pj= (j− 1)mandε= 1的那些；主要的新特征是移位参数(和伴随的Gowers范数对象)到粗尺度和细尺度的局部化，使用正演归纳法将多项式平均线性化，以及在某些代数簇中对有限域上的点数的一些初等估计。
We establish the existence of infinitely manypolynomialprogressions in the primes; more precisely, given any integer-valued polynomialsP1, …,Pk∈Z[m] in one unknownmwithP1(0) = … =Pk(0) = 0, and given anyε> 0, we show that there are infinitely many integersxandm, with, such thatx+P1(m), …,x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear casePj= (j− 1)mandε= 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.