The primes contain arbitrarily long polynomial progressions |
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Authors: | Terence Tao Tamar Ziegler |
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Institution: | 1.Department of Mathematics,University of California, Los Angeles,Los Angeles,U.S.A.;2.Department of Mathematics,Technion – Israel Institute of Technology,Haifa,Israel |
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Abstract: | We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P
1, …, P
k
∈ Zm] in one unknown m with P
1(0) = … = P
k
(0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with
1 \leqslant m \leqslant xe1 \leqslant m \leqslant x^\varepsilon, such that x + P
1(m), …, x + P
k
(m) are simultaneously prime. The arguments are based on those in 18], which treated the linear case P
j
= (j − 1)m and ε = 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. |
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Keywords: | |
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