Year: 2025
Type: article
Abstract: For an integer m greater than 1 $m>1$ m > 1 , we denote by upper P left parenthesis m right parenthesis $P(m)$ P ( m ) the largest prime divisor of m . We improve a result of Stewart [‘On the greatest and least prime factors of n factorial plus 1 ${n!}+1$ n ! + 1 , II’, Publ. Math. Debrecen 65 (3–4) (2004), 461–480] by showing that backslash limsup Underscript n right arrow infinity Endscripts upper P left parenthesis n factorial plus 1 right parenthesis divided by n greater than or slanted equals 1 plus 9 log 2 $\limsup _{n \rightarrow \infty } P({n!}+1)/n \geqslant 1+9\log 2$ \limsup n → ∞ P ( n ! + 1 ) / n ⩾ 1 + 9 log 2 . More generally, for any nonzero polynomial f left parenthesis upper X right parenthesis $f(X)$ f ( X ) with integer coefficients, we show that backslash limsup Underscript n right arrow infinity Endscripts upper P left parenthesis n factorial plus f left parenthesis n right parenthesis right parenthesis divided by n greater than or slanted equals 1 plus 9 log 2 $\limsup _{n \rightarrow \infty } P({n!}+f(n))/n \geqslant 1+9\log 2$ \limsup n → ∞ P (
Author: LI LAI
Institution: Xiamen University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,550
OpenAlex ID: W4415782282