Yeah, but you'd get $1M for disproving it and be remembered as the guy who disproved the Riemann hypothesis. So I don't think there'd be much motivation to keep it secret.

Proving the Riemann hypothesis either way will make you the most famous mathematician in history. Euler would be jealous. (I think I've figured out the proof, but sadly this comment is to short to post it.)

This one works great. Additionally any positive proof of the Riemann hypothesis would probably entail some new deeper understanding of math. A disproof could just consist of a single non-trivial zero with no additional understanding of why it is there.

@quarague not necessarily: it is unlikely the non-trivial zero will just appear as a result of trying to plug in random numbers (if that was possible, I think this would be found already). Therefore, it's more likely that such a zero will not just be sitting there, but rather there will be a description of the process of how that zero was found. Or may be it will be shown that such a zero can be constructed, but not the exact value of it (example: transcendental numbers - for some time we knew they existed, but no one could point out such a number)

@AlmaDo I wrote a disproof could .. give no further insight. Of course it could come with more theory but it doesn't have to.

Isn't pi a transcendental number? It's been known for thousands of years.

@Skyler - the trancendentaility of $\pi$ was only proven within the last 150 years or so (I'm not sure when, but it was not longer than that).

This doesn't force a choice between "Truth" and "prestige" -- being close to the disproof of something this large would "rub off" to a certain extent.

@AlmaDo Of course, that brings up the possibility that the author could have an insightful and constructive proof... but then just use it to figure out a single counterexample and publish that alone without any insight, which could generate interesting conflict in a story. (e.g. the author really wants to spite the mathematical community, who know perfectly well that the counterexample was not found by exhaustive search and would very much like to know the real method)

People have checked that the first 2 million zeroes are on the expected line ... finding a counterexample is not going to be easy.

@AlmaDo: Still a disproof could refrain from disclosing the method by which the zero was found. It would be interesting to see if that happened how long it would take for others to figure out the method.

For added psychological damage, the heroine could prove that the Riemann hypothesis is not right or wrong, but "undecideable": mathoverflow.net/questions/79685/…

Concerning your addendum: isn't that some kind of urban legend? I didn't find any really relevent info of this and I find it hard to believe that Pythagoras thought the square root of two was rational, the proof that it isn't is not that hard, it doesn't involve any clever trick and and I don't see why he wouldn't have seeked it's rational form, since it's a result that often comes up when we use his theorem (every time the triangle has two identical sides in fact)

@Gilsido. The proof involves "proof by contradiction" - was that an accepted basis for proof in Pythagoras's day?

This is a bad idea. Very little in pure math depends on the Riemann Hypothesis alone. The "big conjecture" is the Generalized Riemann Hypothesis, and this has known consequences to estimating the running time of algorithms, so interest in it is not limited to pure math. The OP would be much better off making up a conjecture, loosely inspired by the importance of the Generalized Riemann Hypothesis, instead of having the story's plot depend directly on a real conjecture.

@PaulSinclair: It falls out of $e^{i\pi} = -1$ (and $e$ is transcendental because of Lindemann-Weierstrass)

Possibly useful: this Math.SE thread In the history of mathematics, has there ever been a mistake?. The answer is yes, and you could look into these mistakes to find out what has actually happened in the past.

@MartinBonnersupportsMonica It is very unlikely. The proof by contradiction that √2 is irrational appears first in Aristotle, as an example of proof by contradiction. The formal proof appears in an interpolation in Euclid's Elements.

@Gilsido ""It is only in late sources that the discovery of incommensurability is attributed to the Pythagorean school. The main one is a fragment, surviving in Arabic, of Pappus’ commentary on Book X of the Elements, which says moreover that the first member of the school who divulged the discovery perished in a shipwreck. The same story can be found in a scholium to Book X of the Elements, which perhaps goes back to Proclus. There is also a passage in Proclus, certainly anachronistic, that attributes outright to Pythagoras a theory of irrational numbers."" (books.google.com/?id=ld8lBAAAQBAJ)

@Gilsido Russo writes that Pythagoreans had probably found this problem, because they attached great importance to odd and even numbers and the argument, as you note, it is not difficult to find, but they did not know what to do about it: their whole philosophy was based on integers. The proof by contradiction appears much later, in Plato and Aristotle's time, and the developed theory of real numbers (including irrationals) appears still later, in Euclid, as the theory of ratios of magnitudes.

@AlexP - yes, and Lindemann proved the earliest version of that theorem that is applicable to $\pi$ in 1882, 137 years ago. Before that, no one knew if $\pi$ was transcendental or not.

I am an number theorist, and am not convinced that a disproof of RH would really be that damaging. Sure, most people expect it to be true, and there are a lot of theorems of the form that "RH implies X" but RH isn't really that important in terms of applications. Knowing RH is false (or even independent of ZFC) would just mean that there is something even more interesting going on with the distribution of primes than what we already know.

So, I don't see "lots of damage" neither in your suggestion on what to disprove, nor in your past example in addenum. Scandal among researchers is not "lots of damage".