有加有減階乘機制 - 交錯階乘素數

這跟階乘和函數 (Sum-of-factorials Function) 不同,我們考慮把自 1 起至 n 的階乘 (Factorial) (若不了解什麼是階乘,請到這邊來) 一正一負間隔的加起來,定義交錯階乘 (Alternating Factorial) a(n) 為:

a(n) = n! - (n-1)! + (n-2)! - ..... + (-1)n-k k! + ...

當然我們可以有一簡單的遞推關係式 (Recurrence Relation) 來求交錯階乘:

a(n) = n! - a(n-1) 取 a(1) = 1

因而有數列:0, 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, 1226280710981, 19696509177019, 335990918918981, 6066382786809019, 115578717622022981, 2317323290554617019, 48773618881154822981, ......(OEIS A005165) 。

 

若當中為素數者,便是交錯階乘素數 (Alternating Factorial Prime):

如 5, 19, 101, 619, 4421, 35899, 36614981, ...

數學家相信交錯階乘素數是有限的。而第一個交錯階乘合數 (Alternating Factorial Composite) 為 326981 = 79 * 4139 。

 

參考文獻及網址

Guy, R. K. "Equal Products of Factorials," "Alternating Sums of Factorials," and "Equations Involving Factorial ." §B23, B43, and D25 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 80, 100, and 193-194, 1994.

Weisstein, E. W. "Alternating Factorial." From MathWorld. http://mathworld.wolfram.com/AlternatingFactorial.html.