# Integers such that Difference with Power of 2 is always Prime

## Unproven Hypothesis

The following positive integers are believed to be the only values of $n$ such that:

$\forall k > 0: n - 2^k$

is prime whenever it is (strictly) positive:

$4, 7, 15, 21, 45, 75, 105$

## Examples

### 4

$\forall k > 0: 4 - 2^k$

is prime whenever it is (strictly) positive.

### 7

$\forall k > 0: 7 - 2^k$

is prime whenever it is (strictly) positive.

### 15

$\forall k > 0: 15 - 2^k$

is prime whenever it is (strictly) positive.

### 21

$\forall k > 0: 21 - 2^k$

is prime whenever it is (strictly) positive.

### 45

$\forall k > 0: 45 - 2^k$

is prime whenever it is (strictly) positive.

### 75

$\forall k > 0: 75 - 2^k$

is prime whenever it is (strictly) positive.

### 105

$\forall k > 0: 105 - 2^k$

is prime whenever it is (strictly) positive.

## Historical Note

This conjecture was made by Paul Erdős.