Investigation

## Problem

Consider the two calculations $23=12^2-11^2 \qquad \textrm{and} \qquad 27=14^2-13^2.$

What makes these two calculations interesting to you?

What questions might you ask yourself when considering these two calculations?

Some of your questions may be similar to those below. Investigate the ones that interest you. You might want to make some conjectures and then try to prove (justify) them, or to disprove them by finding counterexamples.

• How many of the numbers from $1$ to $20$ can you express as the difference of two successive perfect squares?

• How many of the numbers from $1$ to $20$ can you express as the difference of two perfect squares?

• How many of the numbers from $1$ to $20$ can you express as the difference of two perfect squares in more than one way?