Let \(\mathcal{P}\) denote the set of primes. For a fixed dimension \(d\), Cook– Magyar–Titichetrakun, Tao–Ziegler and Fox–Zhao independently proved that any subset of positive relative density of \(\mathcal{P}^d\) contains an arbitrary linear configuration. In this paper, we prove that there exists such configuration with the step being a shifted prime (prime minus \(1\) or plus \(1\)).

中文翻译：

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素数中的多维配置具有素数变化的素数阶跃

令\（\ mathcal {P} \）表示素数集。对于固定尺寸\（d \），Cook– Magyar–Titichetrakun，Tao–Ziegler和Fox–Zhao独立证明了\（\ mathcal {P} ^ d \）的正相对密度的任何子集都包含任意线性配置。在本文中，我们证明存在这样的配置，即阶跃是移位的素数（素数减\（1 \）或加\（1 \））。