# Disproof

A statement can be shown to be false by showing that there exists a counter example. For example take the statement $2^p-1$ where $p$ is prime. We can show that this is false by showing one case where this fails. Lets check the cases.

$2^2-1 =3$ is prime

$2^3-1=7$ is prime

$2^5-1= 31$ is prime

$2^7-1=127$ is prime

$2^{11}-1=2047=23 \times 89$ is not prime

Since we found a case that isn’t prime this means that the statement is false. It is generally easier to disprove statements since it only requires one case to be wrong. In order to prove something is accurate it would require a series of accurate logical assertions. However once proven true a statement cannot be disproved. Conversely if a statement is proven false it cannot be proven true.

It is also easier to verify if a proof was done accurately then it is to come up with one. It takes only one small error in one line of of the proof to make it completely invalid. However having an invalid proof doesn’t automatically make the statement false. It could just be that the proof used was inaccurate.

Filed under : Mathematics