# Power of 2 is Difference between Two Powers

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## Theorem

Let $n \in \Z_{>0}$ be a power of $2$.

Then $n$ is the difference between powers of two positive integers greater than or equal to $2$.

## Proof

$2^k = 2^{k+1} - 2^k$

$\blacksquare$

## Examples

### $2^0$ expressed as Difference between Two Powers

- $2^0 = 3^2 - 2^3$

### $2^1$ expressed as Difference between Two Powers

- $2^1 = 3^3 - 5^2$

### $2^2$ expressed as Difference between Two Powers

- $2^2 = 5^3 - 11^2$

### $2^4$ expressed as Difference between Two Powers

- $2^4 = 5^2 - 3^2$

### $2^5$ expressed as Difference between Two Powers

- $2^5 = 3^4 - 7^2$

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $32$