If you are aiming for a career in mathematics at a research university, then trying to complete your Ph.D. as quickly as possible is a bad idea. In the U.S., you should plan on spending four or five years in graduate school. You should not graduate in three or fewer years unless you write an absolutely extraordinary thesis (or personal concerns force you to leave graduate school). Graduating in four years is reasonable, but most people are better off taking five. Even if you've reached a natural break point in your work, it can be valuable to stick around and work on something else for a year.
This may seem counterintuitive, but it's based on three principles that describe how junior candidates are evaluated in mathematics:
The job market is brutally competitive, and you need the strongest application you can put together.
No extra credit is given for graduating quickly. People who graduate in three or four years compete on the same footing as those who spend five. [However, there can be a penalty for taking six or more.]
Once you graduate, the clock starts ticking. If you take longer than average to find a tenure-track job, it will be held against you.
The net result is irrational: the job market judges it better to spend five years in grad school and then do a three-year postdoc than to spend three years in grad school and then five years as a postdoc. The former looks normal, while the latter looks like you graduated too quickly and then were unable to find a tenure-track job after your first postdoc (so what might be considered a compensating plus/minus pair turns into two negatives).
Five years in graduate school is something of a cut-off, for two reasons. Most grad schools discourage staying longer, because of limitations on space and funding, and spending six or more years starts to look like you were unable to complete your dissertation. However, up to five years, longer is usually better.
One way of thinking about this is to consider the question title:
Longer PhD with a deeper result vs a shorter PhD with a sufficient result
What constitutes a "sufficient result"? Sufficient to graduate is a weak condition, since getting a good postdoc is much more difficult than merely completing a dissertation. However, a good postdoc won't get your career off to nearly as strong a start as a great postdoc would. If another year in graduate school could make the difference between a good and a great postdoc, it may well be worth it. But what if your work is so wonderful that you're obviously going to get a great postdoc? At that point you should raise your ambitions and aim to get a Clay research fellowship.
There's an enormously high ceiling, and you never reach the point of being able to say "OK, I've done enough." Graduate school is not a matter of doing enough and seeing how long it takes you. Instead, it's a matter of taking the standard amount of time and seeing how much you can do.