While watching a video about factoring, it occurred to me that, if n is a square number, then n is also the product of square numbers. This is trivially true, of course, in that n is the product of itself and 1, which is a square. If the root of n is not a prime, though, then n can be expressed as a product of squares in at least one other way. To see why this is, consider 36:
36 = (6)(6) = (3)(2)(3)(2) = (3)(3)(2)(2) = (9)(4)
Given any factorization of the root of n, n can be expressed as the product of the squares of the root’s factors.
This is all pretty obvious, but I thought it was interesting. In the future, I may consider if there is a geometric interpretation of this fact and what light it might shed.1
- To do. ↩︎