Consecutive Square Numbers

Question

The difference between the squares of two consecutive numbers is $23$. What are the two numbers?

Answer 1

$a^2-(a+1)^2=2a+1=23$ for some $a \in \textbf{Z}$. Therefore $a=11$.

Answer 2

Let the two numbers be $n$ and $n-1$. The difference of their squares is $$n^2-(n-1)^2 = (n+(n-1))(n-(n-1)) = (2n-1)(1)=2n-1$$ using the difference of two squares factorisation, and this is equal to 23, so: $$2n-1=23$$ giving $$n=12$$ so the numbers are 11 and 12.

Check: $12^2-11^2 = 144-121=23$