设1997＝x，

1997*1998*1999*2000+1

＝x(x+1)(x+2)(x+3)+1
=x^4+6x^311x^2 + 6x + 1
=(x^4 + 3x^3 +x^2) + (3x^3 + 9x^2 +3x) + (x^2 +3x+1)
=x^2(x^2+3x+1) + 3x*(x^2+3x+1) + (x^2+3X+1)
=(x^2+3x+1)^2
= ((x+1)(x+2) -1)^2

带入:

(1998×1999－1)^2

= 15952043988001

1: 1997*1998*1999*2000 +1
2: (1997*2000 ) * (1998*1999) + 1
2.1 1997*2000 = 3994000
2.2: 1998*1999 = 1997*2000 + 2 = 3994002
3: 3994000 * 3994002 + 1
3.1 3994000*3994000 + 3994000*2 + 1
4: ( 3994*3994 ) * 1000 * 1000 + 3994000*2 + 1
4.1: [(4000*3988) + 36] * 1000 * 1000 + 3994000*2 + 1
5 : (15952000 + 36) * 1000 * 1000 + 3994000*2 + 1
6: 15952036,000,000 + 7,988,000 + 1
6.1: 15952043,988,000 + 1
7. 15952043,988,001

set n=2000
1997*1998*1999*2000 +1
= (n-3)(n-2)(n-1)n+1
= ...
= n^4 - 6n^3 + 11n^2 -6n +1
= 16 000 000 000 000 n^4
- 48 000 000 000 6n^3
+ 44 000 000 11n^2
- 12 000 6n
+ 1 1
= 15 952 043 988 001