Friday, 20 September 2019

Fun With Numbers

The Lass and I were brainstorming yesterday, and between us, came up with this sequence/pattern, where x,y are positive integers.

I am using the European convention of using a full stop to denote 'multiply', so "x^3.y" means "x-cubed times y" (to avoid confusion with "x^3y" which could mean "x to the power of three y").

x^1 - y^1 = (x - y)
x^2 - y^2 = (x - y)(x + y)
x^3 - y^3 = (x - y)(x^2 + x.y + y^2)
x^4 - y^4 = (x - y)(x^3 + x^2.y + x.y^2 + y^3)

We didn't work out x^5 - y^5 the long way, but my initial assumption is...
x^5 - y^5 =(x - y)(x^4 + x^3.y + x^2.y^2 + x.y^3 + y^4).

Update 1, PJH in the comments confirms this with link.
Update 2, Bayard and Derek discuss negative powers in the comments. The pattern is fairly straightforward.

For example, assuming x > y, and we want to break down (y^-2) - (x^-2), which can also be expressed as (1/y^2) - (1/x^2).

The answer is a fraction.

The bottom bit is to the power 2, so the top part of the fraction (numerator) is the same as for "x^2 - y^2" in the above list, i.e. "(x - y)(x + y)".

The bottom bit of the fraction (denominator) is just x^2.y^2


Mark Wadsworth said...

There were some good comments on here, unfortunately Blogger had a glitch and deleted them. Sorry all.