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.