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): $127$