Lau 2026 number prime factors consecutive integers

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Cheuk Fung (Joshua) Lau, On the Number of Prime Factors of Consecutive Integers. arXiv preprint (2026). arXiv:2604.15042. The arXiv record (https://arxiv.org/abs/2604.15042, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Theorem 1.1 proves that for some absolute constant C there are infinitely many n with omega(n+k) <= Omega(n+k) <= C log k for all k >= 2, improving Tao and Teräväinen's O(k) bound; Theorem 1.3 gives the same for n-k. Corollary 1.2 deduces tau(n+k) << k^C (a weak form of Erdos #826) and Corollary 1.4 gives omega(n-k) <= k for all large k (a weak form of Erdos #413). The method is a quantitative refinement of the Tao-Teräväinen probabilistic argument: n is drawn from [x,2x] weighted by a product of GPY-type Selberg sieve weights with polynomially decaying sieve levels R_k = x^{c/k^50}, combined with strong exponential concentration for omega under those weights. Conjecture 5 and Conjecture 6, motivated by Cramér-type random models, assert the log k bound is sharp up to a constant, and Section 7 notes that Conjecture 6 would disprove the first part of Erdos #679, which Theorem 1.3 misses by a log log k factor. For Erdos #1203, which asks whether F(n) = max_k omega(n+k) log log k / log k tends to infinity, Theorem 1.1 bounds omega(n+k) by C log k for infinitely many n, a log log k factor above the scale log k / log log k at which F(n) is measured; it neither proves nor refutes that F(n) tends to infinity.

Source: https://arxiv.org/abs/2604.15042.

Bears on. #248, #1203

Results to transcribe.

• Theorem 1.1: There is C > 0 such that for infinitely many n, omega(n+k) <= Omega(n+k) <= C log k for every integer k >= 2.
• Theorem 1.3: Same bound for the descending side: infinitely many n with omega(n-k) <= Omega(n-k) <= C log k for all 1 < k < n.
• Corollary 1.2: There is an absolute C with infinitely many n satisfying tau(n+k) << k^C for all k >= 2 (weak form of Erdos #826).
• Corollary 1.4: There are infinitely many n with omega(n-k) <= k for all sufficiently large k < n (weak form of Erdos #413).
• Conjecture 5 / Conjecture 6: Random-model conjecture that the log k bound is optimal up to a constant; Conjecture 6 would falsify the first claim of Erdos #679.
• Proposition 5.5: Construction of the sieve-weighted random variable n in [x,2x], with p^4 | n for every prime p <= 0.15 log x and upper bounds on the probability that products of distinct primes, and prime powers, divide n+k for 2 <= k <= x^{1/100}; these feed the moment bounds that drive the union bound over k.