Bounds

Upper bounds for every n

Coefficient of (n−3)!	Bound	For	By	Year
926683/18144000.5107	n! + (n-1)! + (n-2)! + \left(926683/1814400 + \varepsilon\right)(n-3)!	large n	The 926683/1814400 bound, Jay Pantone jaypantone	2026	Lean, kernel-checked
1771/34560.5124	n! + (n-1)! + (n-2)! + \left(\tfrac{1771}{3456} + o(1)\right)(n-3)!	large n	The 1771/3456 bound, Jakub Halfar	2026	Lean, kernel-checked
43/800.5375	n! + (n-1)! + (n-2)! + \left(\tfrac{43}{80} + o(1)\right)(n-3)!	large n	The 43/80 bound, Jay Pantone jaypantone	2026	Lean, kernel-checked
3/50.6	n! + (n-1)! + (n-2)! + \tfrac{3}{5}(n-3)!	n ≥ 10	The 3/5 bound, Jay Pantone jaypantone	2026	Lean, kernel-checked
101/1200.8417	n! + (n-1)! + (n-2)! + \tfrac{101}{120}(n-3)!	n ≥ 8	The 101/120 bound, Jay Pantone jaypantone	2026	Lean, kernel-checked
1	n! + (n-1)! + (n-2)! + (n-3)! + n - 4	n ≥ 8	Egan's bound minus one, Uku Raudvere	2026	Lean, kernel-checked
1	n! + (n-1)! + (n-2)! + (n-3)! + n - 3	n ≥ 4	Egan's bound, Greg Egan	2018	Paper
1	1! + 2! + \cdots + n! - 1	n ≥ 6	The recursive bound minus one, Robin Houston	2014	Paper
1	1! + 2! + \cdots + n!	all n	The recursive bound, Daniel A. Ashlock and Jenett Tillotson	1993	Paper

Lower bounds for every n

At n = 13	Bound	For	By	Year
6,746,626,531	F(n) + \left\lceil \frac{2(n-2)! - 2(n-2) - 4p_n/t_n}{n^2 - 4n + 1 - 2(n-1)/t_n} \right\rceil	n ≥ 8	Pantone's bound, Jay Pantone jaypantone	2026	Lean, statement reviewed
6,746,615,766	F(n) + \left\lceil \frac{2((n-2)! - (n-2))}{n^2 - 4n + 1} \right\rceil	n ≥ 7	Halfar's general bound, Jakub Halfar	2026	Lean, statement reviewed
6,746,553,315	F(n) + \left\lceil \frac{2((n-2)! - (n-2))}{n(n-3)} \right\rceil	n ≥ 5	Kisic's bound, Marin Kisic and Jakub Halfar	2026	Lean, kernel-checked
6,746,523,219	H_n + \Gamma_n	n ≥ 5	Liu's bound, Xiaolong Liu	2026	Lean, statement reviewed
6,746,243,919	F(n) + \frac{(n-2)! - (n-2)}{n^2 - 3n + 1}	n ≥ 3	The Hunter–Raudvere bound, Zach Hunter and Uku Raudvere	2026	Lean, statement reviewed
6,746,084,362	F(n) + \left\lceil \frac{(n-3)! - 1}{2n-1} \right\rceil	n ≥ 5	Raudvere's bound, Uku Raudvere	2026	Lean, kernel-checked
6,745,939,211	F(n) + 1	n ≥ 5	4chan + 1, Robin Houston	2018	Lean proof
6,745,939,210	F(n) = n! + (n-1)! + (n-2)! + n - 3	all n	The 4chan bound, Anonymous 4chan poster, Robin Houston, Jay Pantone jaypantone and Vince Vatter	2011	Paper
—	n! + (n-1)! + (n-2)! + \left(2 - \varepsilon\right)(n-4)!	large n	Jay Pantone's bound, Jay Pantone jaypantone	2026	Lean, kernel-checked

Lower bounds for a particular n

n	At least	By	Year
n = 14	93,891,141,150	S(14) ≥ 93,891,141,150, Jay Pantone jaypantone	2026	Lean, kernel-checked
n = 13	6,746,626,957	S(13) ≥ 6,746,626,957, Jakub Halfar	2026	Lean, kernel-checked
n = 12	522,622,380	S(12) ≥ 522,622,380, Jay Pantone jaypantone	2026	Lean, kernel-checked
n = 11	43,917,903	S(11) ≥ 43,917,903, Jakub Halfar	2026	Lean, kernel-checked
n = 10	4,033,379	S(10) ≥ 4,033,379, Jay Pantone jaypantone	2026	Lean, kernel-checked
n = 9	408,469	S(9) ≥ 408,469, Jakub Halfar	2026	Lean, kernel-checked
n = 8	46,133	S(8) ≥ 46,133, Jakub Halfar	2026	Lean, kernel-checked
n = 7	5,898	S(7) ≥ 5,898, Justin Lebar	2026	Lean, native-checked
n = 6	872	S(6) ≥ 872, Benjamin Grayzel	2026	Lean, native-checked
n = 5	153	Exhaustive search, Ben Chaffin	2014	Computer check