Lai 2021 largest prime divisor

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least_prime_divisor_bound: States an improved upper bound for the odd-index liminf of the least prime divisor of n!+1.

lemma_2_7: Bounds a sum of running minima of p-adic valuations over distinct shifted factorial values.

theorem_1_1: Gives a 1+9 log 2 limsup bound, with a positive-lower-density strengthening, for every nonzero polynomial shift of n!.

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Li Lai, On the largest prime divisor of $n!+1$, arXiv:2103.14894v1 (27 March 2021).

The selected arXiv v1 PDF has 11 physical pages. It is a factorial-sequence paper; it does not concern the exponential sequence $2^n-1$ of Problem 977 and is not direct progress on that problem. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2103.14894), every other right reserved.

For every nonzero $f\in\mathbb Z[X]$, Theorem 1.1 proves

$$\limsup_{n\to\infty}\frac{P(n!+f(n))}{n}\geq1+9\log2\approx7.238.$$

More strongly, for each fixed $\varepsilon>0$ the inequality $P(n!+f(n))>(1+9\log2-\varepsilon)n$, together with $n!+f(n)>1$, holds on a set of positive integers $n$ whose lower asymptotic density is positive. This improves the constants $5/2$ of Luca--Shparlinski for general polynomial shifts and $11/2$ of Stewart for $f=1$.

The new ingredient is Lemma 2.7, a simultaneous $p$-adic valuation bound over an ordered family of distinct factorial arguments. The proof of Theorem 1.1 in Section 3 applies that estimate to the prime-power contributions in a large product of values $n!+f(n)$. The introduction also states two further applications of the method: an improved upper bound, approximately $1.293$, for the odd-index liminf of the least prime divisor of $n!+1$, and, on the same p. 2, an improvement of Luca--Shparlinski's lower bound $(2\pi^2+3)/18\approx1.263$ for $\limsup_{n\to\infty}P(n!+2^n-1)/n$ to $1+\frac{2\pi^2-15}{6}\log\frac32\approx1.320$. The paper gives no separate derivation of either application.

Results.

• Theorem 1.1: polynomially shifted factorials
• Lemma 2.7: simultaneous valuation bound
• Unnumbered least-prime-divisor application

The result pages contain exact statements and proof pointers only. No complete proof is transcribed.

Bears on. #977 (context only): Theorem 1.1 with $f=1$ gives $\limsup_{n\to\infty}P(n!+1)/n\geq1+9\log2$ for the factorial variant that the problem's catalog remarks mention; it does not show that $P(n!+1)/n$ tends to infinity and says nothing about $P(2^n-1)/n$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.