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>> 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 thos

>> 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?

kafkaesq 11y ago
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).
- bradleyjg 11y ago
  Thanks for the reply. I assure you I wasn't trolling, I don't know enough about the topic to construct a troll even if I wanted to.
  - kafkaesq 11y ago
    No worries -- your question was totally reasonable (and an in fact, a perfectly good question).
thaumasiotes 11y ago
> 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?
- kafkaesq 11y ago
  I think he just means processing power, generally.

kafkaesq 11y ago

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).

bradleyjg 11y ago
Thanks for the reply. I assure you I wasn't trolling, I don't know enough about the topic to construct a troll even if I wanted to.
- kafkaesq 11y ago
  No worries -- your question was totally reasonable (and an in fact, a perfectly good question).

thaumasiotes 11y ago

> 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?

kafkaesq 11y ago
I think he just means processing power, generally.