Two Simple Number Properties

Reading through Achmad Damar's "104 Number Theory", there are two simple number properties that are mentioned early on that are easy to prove but the proofs are elegant to my eye at least. Let's look at them.

PROPERTY ONE

Let \(n\) be an integer greater than 1. Prove that \(2^n\) is the sum of two consecutive odd integers. 

Let's suppose that the statement is true and that:$$ \begin{align} 2^n &= (2k-1)+(2k+1)\\ \implies \, 2^n &= 4k \\ k &= 2^{n-2}\\ \text{thus } 2^n &= (2^{n-1}-1) + (2^{n-1}+1) \end{align}$$From this we can see how to quickly calculate the two odd consecutive numbers. Suppose \(n=10\) and thus \(2^{10}=1024\). Because \(2^9=512\), it's easy to see that \(1024 = 511 + 513\).

PROPERTY TWO

Let \(n\) be an integer greater than 1. Prove that \(3^n\) is the sum of three consecutive integers.

Again, let's suppose that the statement is true and that:$$ \begin{align} 3^n &= (s-1)+s+(s+1) \\ \implies \, 3^n &= 3s\\ s &= 3^{n-1} \\ \text{thus } 3^n &= (3^{n-1}-1 )+ 3^n + (3^{n-1}+1) \end{align} $$Again it's easy to find these three numbers. Let's take \(n=3\) so that \(3^3=27\) and \(3^2=9\). This gives \(27=8+9+10\).