I know that some famous problems in mathematics have bounties on their heads. (It seems that $1,000,000 is a popular sum.) I was reading about Goldbach's weak conjecture and started thinking whether this is such a great approach. It is known that Goldbach's weak conjecture is true for odd numbers greater than a certain large number. The number is too large to allow checking the remaining cases with a computer. I was thinking that it could be a good idea to set a prize for proving the conjecture and to divide it into portions in a manner of choice, in order to award any progress towards proving the conjecture.

It would be easy to define in this particular case. There is a finite number of numbers left to check and the portion of the prize could be an non-decreasing function of the number of numbers checked. It might be more difficult to find a satisfactory way of dividing a prize for problems like Goldbach's strong conjecture but some ways of doing it can surely be found. I think it could be beneficial, if setting prizes is at all beneficial, which I don't know.

My question is:

Has anything like this been ever done?

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    $\begingroup$ There was the Wolfskehl Prize for FLT, among other things... $\endgroup$ – J. M. isn't a mathematician Feb 10 '12 at 3:21
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    $\begingroup$ I would be very surprised if there were, historically, any prizes for partial progress toward a proof of a big conjecture. The problem is that you can't really tell whether something is an important step towards the final proof, or an interesting dead-end, until an actual proof is found. $\endgroup$ – Cheerful Parsnip Feb 10 '12 at 3:28
  • $\begingroup$ @JimConant I would respectfully disagree. Sometimes progress towards a solution of a theorem can be measured, that is when the set of cases to check is finite. (Or has a finite measure.) It will not be a perfect measurement, because itmay be impossible to know which of those cases are the most difficult, but I don't think it's a serious issue. As I understand, prizes are given with the intention of getting more mathematicians to spend more time on a problem. Similarly, I don't think it's a problem if a prize is given for what turns out to be a dead end. If offering a prize can accelarate the $\endgroup$ – user23211 Feb 10 '12 at 11:59
  • $\begingroup$ realization of the fact that it's a dead end, it's still OK I would believe. $\endgroup$ – user23211 Feb 10 '12 at 12:00
  • $\begingroup$ @ymar: True, true. The classification of finite simple groups was an extreme example of partitioning of a problem into pieces. In my own field of topology, this type of partitioning can rarely be done. $\endgroup$ – Cheerful Parsnip Feb 10 '12 at 12:15

Something like this used to be popular enough back in the nineteenth century (and maybe earlier): various scholarly bodies would pose prize problems and leading mathematicians (and maybe others? history doesn't say so much about that!) would submit "essays", i.e., research papers expressly directed at a particular topic or towards the solution of a particular problem. It is my understanding that they would usually award a prize to at least one person, even if the problem -- if there was a specific problem, which was sometimes not the case -- wasn't completely solved.

The two big examples I can think of off the top of my head are:

1) In 1883 Minkowski (at the age of 18!), together with H. Smith, won a prize set by the French Academy of Sciences for his essay on the theory of quadratic forms.

I should say that I don't know exactly what problem on quadratic forms, if any, the committee had in mind, and Minkowski and Smith's essays do rather different things. Of course, Minkowski's work at least solved a major problem -- the local-global principle for rational quadratic forms -- just maybe not the problem they specifically asked for! (And I think Smith's essay was pretty good too...)

2) In 1887, King Oscar II of Sweden, advised by Mittag-Leffler, established a prize for the solution of the three-body problem. The story of what happened is rather notorious. The following quote is taken from this wikipedia article, which summarizes it well:

In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished Karl Weierstrass, said, "This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics." (The first version of his contribution even contained a serious error; for details see the article by Diacu).

(In fact, if the popular books on the Poincaré Conjecture that I've read can be believed, the parenthetical sentence at the end is an understatement: apparently Poincaré's original paper was something of a mess.) $ $

I do not know of any 20th century or 21st century examples of prize essays like the above, but that doesn't mean they don't exist. The analogue in our day seems to be targeted grants, e.g. the DARPA debacle of a few years ago. But maybe it would be fun to revive the practice of old-timey prize essays? If anyone is reading this who has more money than s/he knows what to do with and aspires to be a philanthropist for mathematics, let me know! In fact, that's a good standing offer, whether you want to fund a prize essay or not. :)

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This is not a direct reply, but topical nontheless. You might be interested to know that not all prizes are 1,000,000 dollars or other large sums. Famously, Paul Erdős used to offer prizes that were proportional to the level of difficulty of the problem (from just a few dollars to thousands of dollars). Some of them are still open (under the management of Ron Graham).

A more recent example: the 289 dollars prize offered by Bill Gasarch on his computational complexity blog was very recently claimed.

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