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

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## 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 $n - 2^k$ is (strictly) positive:

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

This sequence is A039669 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## 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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $105$ - 1994: Richard K. Guy:
*Unsolved Problems in Number Theory*(2nd ed.) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $105$