There is a very nice category of mathematical results (that are also relevant to culture) : the negative results. 

For example there is no solution problem X (Say Fermat last Theorem) is negative and usually the result is not very interesting or motivated for laymen.
But think of the following : There no program checking that a program has no bugs ( by classical diagonal argument: how would this program test itself). 
In this case we prove something is impossible and so we save a hell of a lot of time: no need to search any more. 
Negative results are extremely useful : in a negative way you avoid loosing money and in fact it is a good justification for pure research. 

Yet beware that tough simple negative statements are not always understood some people say : "Oh Fermat equation has no solution , that is because they did not try hard enough , I will do it". 
This is akin to the trisectors and other poor souls looking for perpetual motion machines.