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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.

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    $\begingroup$ How about Goldbach? $\endgroup$ – carmichael561 Sep 23 '16 at 23:01
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    $\begingroup$ Lothar Collatz seems a good candidate, I'd say $\endgroup$ – sTertooy Sep 23 '16 at 23:01
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    $\begingroup$ To my perception, this is one of those misleading issues where we are aware of a certain number of positives, but/and unaware of negatives so we can't talk about them usefully. So various dim historical figures, despite making no definite mathematical contribution, made (by accident?) a conjecture that not only has defied resolution but may seem to have significance. My point is that making a single such is probably of no consequence... but making several (at least two?), starts to become interesting, perhaps? But I don't know of such, apart from mathematicians who have larger careers... $\endgroup$ – paul garrett Sep 23 '16 at 23:11
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    $\begingroup$ Not that it really matters, but should there be criteria for famous among whom? de Fermat probably is most famous for his conjecture, that is I suspect that among all people who recognise the name at all, a majority of them couldn't say a single other thing he's associated with because nothing else ever made the news. Whereas a minority of people who recognise the name, including essentially all mathematicians, will at least be able to offer you the Little Theorem. FWIW, Wikipedia concurs, "He is best known for Fermat's Last Theorem". $\endgroup$ – Steve Jessop Sep 24 '16 at 8:28
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    $\begingroup$ Would Ramanujan qualify? He discovered a great many results, but the task of giving formal proofs usually fell to others. $\endgroup$ – Dave Radcliffe Sep 24 '16 at 19:22
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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.

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    $\begingroup$ Wow a meta-answer actually serves as a sort-of answer on Main SE. $\endgroup$ – user21820 Sep 24 '16 at 15:12
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    $\begingroup$ Good choice, convincingly justified. Thanks! $\endgroup$ – Joseph O'Rourke Sep 24 '16 at 18:30
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Andrew Beal, while not strictly a mathematician, has not proven (to my knowledge) any mathematical result, despite having a rather famous unsolved problem in his name with a monetary prize that is the same of that of any Clay Mathematics prize.

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    $\begingroup$ I feel that I should mention that there's a large prize associated with Beal's conjecture not because the result is interesting or would lead to interesting mathematics, but because Beal is a billionaire. $\endgroup$ – anomaly Sep 23 '16 at 23:22
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    $\begingroup$ @anomaly Yes, but the goal of the prize was to get young people interested in mathematics. It's not like he just decided to throw his money at random stuff. $\endgroup$ – Carl Schildkraut Sep 23 '16 at 23:23
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    $\begingroup$ Yeah, but he threw money at the conjecture because it was his own (though probably not original to him). It's not like there was a huge interest or need among mathematicians for this kind of mostly-recreational number theory that Beal was trying to promote. (The 'mostly' is there because it follows (modulo a finite number of countrexamples) from the abc conjecture. It's also vaguely related to FLT, which is interesting because of the Taniyama–Shimura–Weil conjecture.) He's a rich guy throwing money at a problem he came up with. That's fine, but the problem itself is unremarkable. $\endgroup$ – anomaly Sep 23 '16 at 23:26
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    $\begingroup$ Interesting & clever suggestion. But, as you acknowledge, Beal is not "strictly" a mathematician. $\endgroup$ – Joseph O'Rourke Sep 24 '16 at 0:51
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Franz Mertens was a late 19th/early 20th century German mathematician who is known for some results about the density of prime numbers and even has a constant named after him. But what he is probably most famous for, no doubt because of the elementary nature of the statement, is the Mertens conjecture, which stated:

Mertens' conjecture: For all $n>1,$ $$\left\lvert \sum_{k=1}^{n}\mu(k) \right\rvert < \sqrt{n},$$ where $\mu\colon\mathbb{N}\to\{-1,0,1\}$ is the Möbius function.

This conjecture, if true, would imply the Riemann hypothesis (!) - hence its fame. Unfortunately, in spite of tremendous amounts of numerical evidence in favour, it was proved false in the 1980s. The smallest counterexample is known to be larger than $10^{14},$ but the upper bound on it is truly enormous: $\operatorname{exp}{(1.59\times10^{40})}.$

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As others have pointed out, "famous" is subjective and a bit ephemeral. I would like to adjust the question to ask for mathematicians who have actually contributed more to mathematics by their conjectures than via their theorems. This is still subjective, but distinctly less so.

Anyway, when I try to think of a mathematician whose greatest contributions lie with his conjectures and for which those contributions are enormous, one name springs to mind: Robert Langlands. The Langlands program is one of the most important and influential pieces of 20th and 21st century mathematics. I would imagine that Langlands himself would agree that his program is (even) more important than the results he has proved: in fact, most or all of his important results feed into and elevate his program.

The OP says

I am interested to explore whether some mathematicians have specific conjecture-talent that is not evidently reflected in theorem-proving talent.

I think this is a good example of this, of a certain particular kind: Langlands clearly has remarkable theorem-proving talent. However his conjecture-talent is beyond remarkable...it is Langlandsesque.

I think there is a further small perturbation of the question which makes the canonical answer Paul Erdős. If we rank mathematicians by number of theorems proved then Erdős surely comes near the top of the list. However, the influence that he has had on contemporary mathematics goes beyond any one result of his. Erdős died (almost exactly) 20 years ago. That was right about the point where I started paying attention to the mathematical landscape, in particular number theory. The last 20 years have been a stampede towards the kind of problems that Erdős proposed. In particular, the amount of leading mathematical work done in that time towards the Erdős–Turán conjecture alone is enormous.

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  • $\begingroup$ "...it is Langlandsesque" :-) $\endgroup$ – Joseph O'Rourke Sep 25 '16 at 19:47
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Mikhail Yakovlevich Suslin made some important discoveries, and no doubt would have made many more had he not died tragically young, but he is most famous for Suslin's problem. It is true that he stated it as a question rather than a conjecture, but I'm not sure if you are insisting on that distinction. Rightly or wrongly it's often called Suslin's hypothesis; the difference between a hypothesis and a conjecture seems kind of nebulous to me.

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    $\begingroup$ I assume it's called a hypothesis because, like the continuum hypothesis, it is independent of ZFC, and so its interest lies in the mathematics that comes from assuming it or its negation as a hypothesis. (One can say the same thing about the Riemann hypothesis, except that it is only possibly independent as yet.) $\endgroup$ – Mario Carneiro Sep 25 '16 at 8:13
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Taniyama-Shimura-Weil conjecture which states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles used this conjecture to establish the modularity theorem for semistable elliptic curve. This became the basis of Wiles proof of Fermat's last theorem. Yutaka Taniyama never lived to see the fruits of his work as he had committed suicide at a young age. Taniyama once remarked that he himself did not fully understand the conjecture and many mathematicians thought that it was a tough conjecture to work upon.

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    $\begingroup$ (I think it was a surprising result that in every case they could check, it was always the L-function of a weight $2$ modular form, so it had to be a theorem behind this, even in the case the general conjecture was false) $\endgroup$ – reuns Sep 24 '16 at 6:02
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Most of the people who have famous conjectures would be pretty well-known even without those conjectures, but maybe these are some exceptions (from Number Theory): Giuga; Catalan; Bertrand; Waring.

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    $\begingroup$ Is Catalan's conjecture better known or more relevant than his numbers? $\endgroup$ – quid♦ Sep 24 '16 at 15:05
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    $\begingroup$ @quid, that's a tough call. My subjective opinion is that he's better known for his conjecture than for his numbers, at least in the Number Theory community, but I concede it could go either way. $\endgroup$ – Gerry Myerson Sep 25 '16 at 0:08
  • $\begingroup$ But among combinatorialists I'd imagine he's better known for his numbers. $\endgroup$ – Michael Lugo Mar 28 '17 at 17:17
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Bernhard Riemann.

Riemann died shortly before his $40$th birthday.

Lots of things are named after him: see here.

However, the most famous one is probably the Riemann hypothesis.

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    $\begingroup$ (together with : Riemann integral, Riemann surface, Riemannian manifold, Riemann sphere) $\endgroup$ – reuns Sep 24 '16 at 0:33
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    $\begingroup$ Cauchy–Riemann equations, Riemann mapping theorem, Riemann–Roch theorem$\,\ldots\qquad$ $\endgroup$ – Michael Hardy Sep 24 '16 at 0:39
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    $\begingroup$ On the one hand, I agree with this answer because Riemann is more famous for his conjectures than his theorems: any layperson who has heard of the Riemann hypothesis is a piece of evidence towards this, and there are plenty such laypersons. If a layperson knows about, say, the Cauchy-Riemann equations, then I would say that they aren't a layperson. But on the other hand, Fermat has been decreed invalid by the OP because "he established so many results independent of his conjecture," and this is quite sensible really since, without this criterion, Fermat and Riemann are obvious answers. $\endgroup$ – Will R Sep 24 '16 at 15:22
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    $\begingroup$ In the same vein we could ask: Which mathematicians are most famous for their lemmas? (Zorn?) Axioms? (Zermelo, Fraenkel?) Paradoxes? (Russell?) Definitions? $\endgroup$ – bof Sep 25 '16 at 0:33
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    $\begingroup$ I've heard of Zorn's lemma, the Peano axioms, Matijasevich's theorem, Euclid's algorithm, Maxwell's equations, Ptolemy's table, Euler's identity, Skewes' number, Sierpiński's constant, Hilbert's problems, Riemann's zeta function, Goldbach's conjecture, Lewy's example, Furstenberg's proof, Newton's method, Cantor's argument, etc. And for each of those there are others with the same common noun but a different proper name. But what about "definition"? Is "Tarski's definition" the only one? $\qquad$ $\endgroup$ – Michael Hardy Sep 25 '16 at 4:16

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