Chen 2022 conjecture erdos

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Yong-Gao Chen, Yuchen Ding, On a conjecture of Erdős. arXiv:2201.10727 (2022); published as Comptes Rendus Mathématique 360 (2022), 971--974, doi 10.5802/crmath.345, online 2022-09-29. The arXiv record (https://arxiv.org/abs/2201.10727, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Theorem 1.1 shows that for any l distinct integers a_1, ..., a_l there are infinitely many n for which the number of solutions of n = p + a_i with p prime exceeds (1/8) log l - 1.6. Corollary 1.2 deduces that if a_1 < ... < a_t <= x with t > log x then infinitely many n have more than (1/8) log log x - 1.6 such representations, which confirms Erdos's 1950 conjecture that this count exceeds any fixed constant c for large x; Corollary 1.3 gives lim sup f_A(n) = infinity for every infinite set A. The proof is short and rests on the Maynard-Tao theorem (Lemma 2.4, cited from Granville's survey, Theorem 6.2: for an integer m >= 2 and k with k log k > e^{8m+4}, every admissible k-set has infinitely many translates containing at least m primes), together with Chen and Sun's explicit Mertens-type bound prod_{3 <= p <= x} (1 - 1/p)^{-1} <= 0.923 log x for x >= 74 (Lemma 2.5), used to extract from any l distinct integers a large admissible subset by successively discarding one residue class modulo each small prime. Erdos had proved the special case a_i = 2^i, and earlier partial cases (a_i | a_{i+1}, and a_i = 2^{p_i} with p_i the i-th prime) were handled by Ding and by Ding and Zhou. For problem 237, which is this conjecture of Erdos, the paper gives the full resolution with a quantitative log log x lower bound.

Source: https://arxiv.org/abs/2201.10727.

Bears on. #237

Results to transcribe.

• Theorem 1.1: For any l distinct integers a_1,...,a_l, infinitely many n have more than (1/8) log l - 1.6 representations n = p + a_i with p prime.
• Corollary 1.2: If x >= 2 and a_1 < ... < a_t <= x with t > log x, infinitely many n have more than (1/8) log log x - 1.6 such representations, confirming Erdos's conjecture.
• Corollary 1.3: For any infinite set A of integers, lim sup_n #{(p,a) : n = p + a} = infinity.
• Method: Extract a large admissible subset by removing one residue class per small prime, then apply the Maynard-Tao bounded-gaps machinery.