Power of 2 is Difference between Two Powers
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The validity of the material on this page is questionable.
This is so trivial I wonder whether something got lost in translation.
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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 $2$.
This is so trivial I wonder whether something got lost in translation.
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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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $32$