It's a big world, not everyone needs to be a farmer/doctor/psychologist so some people who have other skills do other things. Sometimes those other things help. A lot of time they don't. For better or worse we live in a society.

> ... but some things have an impact on people's quality of life: nutrition, healthcare, and psychological well-being.

I would perhaps classify that as "immediate" impact, not "lack of impact".

Number theory has driven computer science, calculus, statistics, and so on.

CS, calculus, stats... they all drive (for example) "nutrition". For example, how do you know what a "vitamin" is? How do you measure it? How do you quantify it's effects on the body (and mind)? A whackload of analysis DEPENDS on number theory for us to understand almost everything about the natural world!

Psychological well being is definitely improved if people could work on something they like and get praised/rewarded for it. And I believe that is way people could get interested in something. Best engineers I know of, had got into their field because they are driven by solving problem just for the sake of it.

I'd say there's a strong case that this has a positive impact on the psychological well-being of anyone who's interested in number theory.

Perhaps not that many people in the world understand the formulation of the problem, but then not many people "get" abstract art, either.

So? What is the point of living?

:-)