Claims

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2004_04_08_green_tao: Theorem 1.2 of Green and Tao (Ann. of Math. 2008) gives a color class of any finite coloring k primes in progression, answering the first question yes; refereed; the first question only.

2026_04_08_bloom: The site curator's own counterexample to the second question: coloring the integers by residue modulo 4 leaves no monochromatic pair with prime difference; no publication or independent review exists, so it is pending.

2026_05_12_kitamura: Kenta Kitamura's Lean 4 development of 12 May 2026, prepared with Codex and GPT-5.5 xhigh, proves that coloring the natural numbers modulo 4 leaves no monochromatic progression with prime difference; pending, second question only.

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