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 $n - 2^k$ 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): $105$