Power of 2 is Difference between Two Powers

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 $2$.

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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

• 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)

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