表方幂为连续的阶乘和(英文)

Representation of a Power As a Sum of Consecutive Factorials

考虑方程am=n!+(n+1)!+……+(n+k)!,其中a>1,m>1,n≥1.我们证明了当a 0 mod 223092870时,方程所有的解是23=2!+3!,32=1!+2!+3!,25=2!+3!+4!,122=4!+5!;当a≡0 mod 223092870时,令p是满足p a的最小素数,如果方程有解,则m≤p.而且,我们猜想上述的四个解是方程仅有的解.

Consider the equation a^m = n ! ＋ （ n ＋ 1 ） ! ＋ ... ＋ （ n ＋ k） ! with a 〉 1, m 〉 1, n≥ 1. We show that when a≠0 mod 223092870, all solutions of the equation are 2^3 = 2 ! ＋ 3 ! ,3^2 = 1 ! ＋ 2 ! ＋ 3 ! ,2^5 = 2 ! ＋ 3 ! ＋ 4 ! and 12^2 = 4 ! ＋ 5 ! ; when a=0 mod 223092870, let p be the least prime with p=a, if the equation has solutions, then m≤p. Moreover, we conjecture that the foregoing four solutions are only solutions of the equation.