Integers such that Difference with Power of 2 is always Prime/Examples/45
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Example of Integers such that Difference with Power of 2 is always Prime
The positive integer $45$ has the property that such that:
- $\forall k > 0: 45 - 2^k$
is prime whenever it is (strictly) positive.
Proof
\(\ds 45 - 2^1\) | \(=\) | \(\ds 43\) | which is prime | |||||||||||
\(\ds 45 - 2^2\) | \(=\) | \(\ds 41\) | which is prime | |||||||||||
\(\ds 45 - 2^3\) | \(=\) | \(\ds 37\) | which is prime | |||||||||||
\(\ds 45 - 2^4\) | \(=\) | \(\ds 29\) | which is prime | |||||||||||
\(\ds 45 - 2^5\) | \(=\) | \(\ds 13\) | which is prime | |||||||||||
\(\ds 45 - 2^6\) | \(=\) | \(\ds -19\) | which is not positive |
$\blacksquare$