An improvement on the largest prime factors of consecutive integers

2026/07/17 by Zhiyuan Yang
#math.NT

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Abstract

Let P⁺(n) denote the largest prime factor of n. One of Erdős and Turán's conjectures asserts that the asymptotic density of integers n satisfying P⁺(n)<P⁺(n+1) is 1/2. In this paper, we prove that this density is larger than 0.280, which improves the previous result 0.2017 by Lü and Wang (2025). We also prove that there exists a positive density of n such that P⁺(n)<P⁺(n+1)<x^41/107+ε. Define T_c(x):=#\p≤ x:P⁺(p-1)≥ p^c\. For 1/2<c<1, we also show that \mathoplim sup_x→∞(T_c(x))/(π(x))≤ min(-(7)/(2)log c,(1-δ)/(2c)), where δ=δ(c)>0.

Citations

• Primes in arithmetic progressions to large moduli

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