# What conjectures are there concerning differences of squares?

May 14, 2017

${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

#### Explanation:

${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

We can prove this by simply expanding $\left(a + b\right) \left(a - b\right)$:
$\left(a + b\right) \left(a - b\right)$
$= {a}^{2} - a b + a b - {b}^{2}$
$= {a}^{2} - {b}^{2}$

May 14, 2017

A few thoughts...

#### Explanation:

Most properties of differences of squares seem to be fairly straightforward to prove. I suppose if you did not know the proofs then they would be conjectures. For example:

• Any odd number is the difference of squares of two integers.

• Any multiple of $4$ is the difference of squares of two integers.

• Any integer of the form $4 k + 2$ (where $k$ is an integer) is not the difference of squares of two integers.