De Polignac's False Conjecture

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

$\blacksquare$

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.