N=1/5*n^5+1/3*n^3+7/15*n
=1/15*(3n^5+5n^3+7n)
=1/15*n(3n^4+5n^2+7)
=1/15*n(3n^4-10n^2+7+15n^2)
=n/15*[(3n^2-7)(n^2-1)+15n^2]
=(n-1)n(n+1)(3n^2-7)/15+n^3
因为(n-1),n,(n+1)是3个连续的自然数,一定有个是3的倍数.
如果(n-1),n,(n+1)里有5的因子,则(n-1)n(n+1)是15的倍数,得证.
如果(n-1),n,(n+1)里没有5的因子,
则只能是n=5k+2,n=5k+3.(k是自然数)
n=5k+2时,
3n^2-7=3(5k+2)^2-7=75k^2+60k+5,是5的倍数.
n=5k+3时,
3n^2-7=3(5k+3)^2-7=75k^2+90k+20,是5的倍数.
所以(n-1)n(n+1)(3n^2-7)是15的倍数.
得证.