There is a very nice category of mathematical results that are relevant to culture ; the negative results. Like there is no solution to such and such
Say Fermat last Theorem is negative but no 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 never ending machines.