I'm wondering if some mathematicians gained more fame from their (perhaps visionary) conjectures, than from the positive results they proved?
I would say this is not true of Fermat, despite his famous eponymous conjecture
(now settled),
because he established so many results independent of his conjecture.
And this seems not true of Poincaré, whose famous conjecture (also now settled) bears his name.
But he was incredibly accomplished independent of that conjecture.
Atiyah has formulated wonderfully productive conjectures (one leading to a Witten advance), but he has also
established major results, e.g., the Atiyah-Singer theorem.
I am interested to explore whether some mathematicians have specific conjecture-talent that is not evidently reflected in theorem-proving talent.
Best Answer
Lothar Collatz: while he is a celebrated mathematician who has a formula named after him (Collatz-Wielandt formula) and he has received quite a few honorary degrees (see wiki), he is definitely best known for his Collatz conjecture, also known as the $3n+1$ conjecture, which he posed in 1937.
It remains unsolved up to this day, despite numerous attempt by professional and amateur mathematicians, and its popularity can be seen even here. In fact, I would argue that the best "proof" that his conjecture is more famous than his actual work, is that the tag (collatz) refers exclusively to the Collatz conjecture.