# De Polignac's False Conjecture/Investigation

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## De Polignac's False Conjecture

It is seen by direct investigation that the first few integers support the conjecture.

As follows:

\(\ds 3\) | \(=\) | \(\ds 2^0 + 2\) | ||||||||||||

\(\ds 5\) | \(=\) | \(\ds 2^1 + 3\) | ||||||||||||

\(\ds 7\) | \(=\) | \(\ds 2^2 + 3\) | ||||||||||||

\(\ds 9\) | \(=\) | \(\ds 2^1 + 7\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^2 + 5\) | ||||||||||||

\(\ds 11\) | \(=\) | \(\ds 2^2 + 7\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 3\) | ||||||||||||

\(\ds 13\) | \(=\) | \(\ds 2^1 + 11\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 5\) | ||||||||||||

\(\ds 15\) | \(=\) | \(\ds 2^1 + 13\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^2 + 11\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 7\) | ||||||||||||

\(\ds 17\) | \(=\) | \(\ds 2^2 + 13\) | ||||||||||||

\(\ds 19\) | \(=\) | \(\ds 2^1 + 17\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 11\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 3\) | ||||||||||||

\(\ds 21\) | \(=\) | \(\ds 2^1 + 19\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^2 + 17\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 13\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 5\) | ||||||||||||

\(\ds 23\) | \(=\) | \(\ds 2^2 + 19\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 7\) | ||||||||||||

\(\ds 25\) | \(=\) | \(\ds 2^1 + 23\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 17\) | ||||||||||||

\(\ds 27\) | \(=\) | \(\ds 2^3 + 19\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 11\) | ||||||||||||

\(\ds 29\) | \(=\) | \(\ds 2^4 + 13\) | ||||||||||||

\(\ds 31\) | \(=\) | \(\ds 2^1 + 29\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 23\) | ||||||||||||

\(\ds 33\) | \(=\) | \(\ds 2^1 + 31\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^2 + 29\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 17\) | ||||||||||||

\(\ds 35\) | \(=\) | \(\ds 2^2 + 31\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 19\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 3\) | ||||||||||||

\(\ds 37\) | \(=\) | \(\ds 2^3 + 29\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 5\) | ||||||||||||

\(\ds 39\) | \(=\) | \(\ds 2^1 + 37\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 31\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 23\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 7\) | ||||||||||||

\(\ds 41\) | \(=\) | \(\ds 2^2 + 37\) | ||||||||||||

\(\ds 43\) | \(=\) | \(\ds 2^1 + 41\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 11\) | ||||||||||||

\(\ds 45\) | \(=\) | \(\ds 2^1 + 43\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^2 + 41\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 37\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 29\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 13\) | ||||||||||||

\(\ds 47\) | \(=\) | \(\ds 2^2 + 43\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 31\) | ||||||||||||

\(\ds 49\) | \(=\) | \(\ds 2^1 + 47\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 41\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 17\) | ||||||||||||

\(\ds 51\) | \(=\) | \(\ds 2^2 + 47\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 43\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 19\) | ||||||||||||

\(\ds 53\) | \(=\) | \(\ds 2^4 + 37\) | ||||||||||||

\(\ds 55\) | \(=\) | \(\ds 2^1 + 53\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 47\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 23\) | ||||||||||||

\(\ds 57\) | \(=\) | \(\ds 2^2 + 53\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 41\) | ||||||||||||

\(\ds 59\) | \(=\) | \(\ds 2^4 + 43\) | ||||||||||||

\(\ds 61\) | \(=\) | \(\ds 2^1 + 59\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^3 + 53\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 29\) | ||||||||||||

\(\ds 63\) | \(=\) | \(\ds 2^1 + 61\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^2 + 59\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^4 + 47\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^5 + 31\) | ||||||||||||

\(\ds 65\) | \(=\) | \(\ds 2^2 + 61\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2^6 + 1\) |

$\blacksquare$