I'm a mathematician and I'd say their ideas only become interesting if that conjecture is true. Otherwise it's just numerology.
To me, this makes for a very boring notion of "interesting."
I think most mathematicians would say that the "interestingness" of a conjecture comes from it (1) describing a phenomenon which seems "intuitively true, or very likely true" (e.g. "x^n+y^n=z^n has no solutions for n>2") combined with (2) the initial difficulty of deciding its truth/falsity using tools available at the time of its statement; along with, finally: (3) the novel techniques (sometime first arising in our brains decades or generations later!) required to ultimately determine said truth/falsity (and the degree to which these techniques touch on and illuminate other areas of mathematics).
For example, I think you'd find near-universal agreement among mathematicians that not only would the resolution of FLT (as a conjecture stated my Fermat) would have been equally "interesting" if it had been proven false -- it may have even been more surprising if a counter-example had been found (or its existence proven), provided the tools / lessons were as interesting as those in the Taylor-Wiles result we know today.
Meanwhile, some the most interesting conjectures are perhaps those that can't be decided, one way or another.
EDIT: If you don't like the idea of discussion the "what-ifs" of a conjecture that's already been decided (like FLT), just plug in any of the usual suspects, e.g. RH or GRH into what I'm saying above. Clearly, a "false" determination on any of these of these major targets -- or even a serious hint at it -- would be career-making achievement for an aspiring mathematician.
>> it may have even been more surprising if a counter-example had been found (or its existence proven), provided the tools / lessons were as interesting as those in the Taylor-Wiles result we know today.
What if a counterexample with very large (x,y,z,n) had been found somewhere in the late 1980s because enough megaflops to find it was finally allocated the problem? Would that necessarily have been an interesting result?
> What if a counterexample with very large (x,y,z,n) had been found somewhere in the late 1980s because enough megaflops to find it was finally allocated
I'm curious why floating point operations would be the appropriate tool for finding solutions to a Diophantine equation. Did you have something in mind when writing this? Where can I learn more about it?
Not sure how to answer you on this (because there's a slight chance you might be trolling). Let's just say superficially "yes", in that it would mean the current expert consensus in our universe (that FLT has been proven) would have to be wrong.
But it's kind of a bad line of speculation (and so FLT probably wasn't the best illustrative example to bring up in my original post); again, for a real-life instance of a counter-example being found to a conjecture that had a lot of numerical evidence suggestion there wouldn't be one, have a look a the history of the Mertens Conjecture, and others of its ilk.
Basic point being that yes, counter-examples to interesting conjectures are always interesting results (and by themselves don't make the original conjecture any less interesting).
To me, this makes for a very boring notion of "interesting."
I think most mathematicians would say that the "interestingness" of a conjecture comes from it (1) describing a phenomenon which seems "intuitively true, or very likely true" (e.g. "x^n+y^n=z^n has no solutions for n>2") combined with (2) the initial difficulty of deciding its truth/falsity using tools available at the time of its statement; along with, finally: (3) the novel techniques (sometime first arising in our brains decades or generations later!) required to ultimately determine said truth/falsity (and the degree to which these techniques touch on and illuminate other areas of mathematics).
For example, I think you'd find near-universal agreement among mathematicians that not only would the resolution of FLT (as a conjecture stated my Fermat) would have been equally "interesting" if it had been proven false -- it may have even been more surprising if a counter-example had been found (or its existence proven), provided the tools / lessons were as interesting as those in the Taylor-Wiles result we know today.
Meanwhile, some the most interesting conjectures are perhaps those that can't be decided, one way or another.
EDIT: If you don't like the idea of discussion the "what-ifs" of a conjecture that's already been decided (like FLT), just plug in any of the usual suspects, e.g. RH or GRH into what I'm saying above. Clearly, a "false" determination on any of these of these major targets -- or even a serious hint at it -- would be career-making achievement for an aspiring mathematician.