# De Polignac's False Conjecture

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## Famous False Conjecture

Every odd number greater than $1$ can be expressed as the sum of a power of $2$ and a prime.

### Investigation

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

## Refutation

The smallest integer for which this conjecture fails is $127$:

\(\ds 127 - 2^0\) | \(=\) | \(\, \ds 126 \, \) | \(\, \ds = \, \) | \(\ds 2 \times 3^2 \times 7\) | not prime | |||||||||

\(\ds 127 - 2^1\) | \(=\) | \(\, \ds 125 \, \) | \(\, \ds = \, \) | \(\ds 5^3\) | not prime | |||||||||

\(\ds 127 - 2^2\) | \(=\) | \(\, \ds 123 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 41\) | not prime | |||||||||

\(\ds 127 - 2^3\) | \(=\) | \(\, \ds 119 \, \) | \(\, \ds = \, \) | \(\ds 7 \times 17\) | not prime | |||||||||

\(\ds 127 - 2^4\) | \(=\) | \(\, \ds 111 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 37\) | not prime | |||||||||

\(\ds 127 - 2^5\) | \(=\) | \(\, \ds 95 \, \) | \(\, \ds = \, \) | \(\ds 5 \times 19\) | not prime | |||||||||

\(\ds 127 - 2^6\) | \(=\) | \(\, \ds 63 \, \) | \(\, \ds = \, \) | \(\ds 3^2 \times 7\) | not prime | |||||||||

\(\ds 127 - 2^7\) | \(=\) | \(\, \ds -1 \, \) | \(\ds \) | and we have fallen off the end |

$\blacksquare$

## Also see

## Source of Name

This entry was named for Alphonse de Polignac.

## Historical Note

Alphonse de Polignac proposed his false conjecture in $1848$, claiming that he had verified it up to $3$ million.

This was clearly an exaggeration, as the conjecture fails for the modestly small number $127$.

David Wells, in his *Curious and Interesting Numbers* of $1986$, mentions that this topic is discussed by Nigel Boston in *Quarch*, no. $6$.

This needs to be corroborated.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $127$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $127$