Show that the product of three consecutive positive integers cannot be the n-th power of an integer, for any integer n>1.

(In reply to re(2): Solution? by KC)

Yes, you are correct.  Each prime factor of b^n must occur a multiple of n times.  Hence if b^n=uv, with u relatively prime to v, both u and v are n-th powers.

Also, if anyone asks, consecutive n-th powers differ by more than n because, for z>=1,

(z+1)^n-z^n=z^n+nz^(n-1)+positive stuff-z^n>nz^(n-1)>=n.

Posted by McWorter on 2005-09-05 22:04:49