5 ms·

1e+33 is not such a big number as far as number theory goes (for example: http://mathoverflow.net/questions/15444/examples-of-eventual-counterexamples http://ma

1e+33 is not such a big number as far as number theory goes (for example: http://mathoverflow.net/questions/15444/examples-of-eventual-counterexamples http://mathoverflow.net/questions/15444/examples-of-eventual..., https://en.wikipedia.org/wiki/Skewes%27_number https://en.wikipedia.org/wiki/Skewes%27_number). It's still a nice exercise. As another example, consider the question of whether there exists a right-angled triangle with rational sides, having an area of 157.

acqq 10y ago
If I understand it right, it's a number with which he could have estimated if it's worth using a computer to perform a brute force search for a possible solution. Once he had 1e-33 for all n > 100 that could mean that even trying with 1M computers where each tries 10G solutions per second (1e6*1e10) some millions of years could pass without the positive result. Then it's exactly reasonable to say "for my money Fermat’s theorem is true" as in, really not worth trying blindly.
- conistonwater 10y ago
  I can't believe that the right interpretation of what he said. He must have known what the actual FLT meant. Furthermore, it would go against the mathematical tradition of what it means to "think that a conjecture is true". Compare with RH, for example: when people say it's probably true they absolutely do not mean it's true for all small numbers, in part because there are built-up areas of mathematics that depend on it being exactly true. Besides, brute-force search is a pretty terrible algorithm (in general), so finding out that it fails on a particular problem isn't that interesting.
  - acqq 10y ago
    He died in 1988. Imagine somebody came to you at these "early" times (Wiles proved the theorem in 1995) with a "grand project" to use a lot of computers to search for a possible "solution," being able to try 10 billion Ns in one second. What would be your argument against such a project? Would you try some similar derivation to get an estimate of success? I know, brute forcing all integers is impossible, but you can even "imagine" a "superquantum (and now not existing) computer" which can do a lot of small integers in parallel. And yes, I know, probabilities aren't proofs, especially not for integers. One counterexample is enough: 3987^{12} + 4365^{12} = 4472^{12} Also interesting to see the accidental(?) order of magnitude of the numbers involved.
talaketu 10y ago
Wasn't the idea that he integrated on n for 100 to infinity, to arrive at that probability? So the probably should be interpreted as the probability of an eventual counter-example.

acqq 10y ago

If I understand it right, it's a number with which he could have estimated if it's worth using a computer to perform a brute force search for a possible solution. Once he had 1e-33 for all n > 100 that could mean that even trying with 1M computers where each tries 10G solutions per second (1e6*1e10) some millions of years could pass without the positive result. Then it's exactly reasonable to say "for my money Fermat’s theorem is true" as in, really not worth trying blindly.

conistonwater 10y ago
I can't believe that the right interpretation of what he said. He must have known what the actual FLT meant. Furthermore, it would go against the mathematical tradition of what it means to "think that a conjecture is true". Compare with RH, for example: when people say it's probably true they absolutely do not mean it's true for all small numbers, in part because there are built-up areas of mathematics that depend on it being exactly true. Besides, brute-force search is a pretty terrible algorithm (in general), so finding out that it fails on a particular problem isn't that interesting.
- acqq 10y ago
  He died in 1988. Imagine somebody came to you at these "early" times (Wiles proved the theorem in 1995) with a "grand project" to use a lot of computers to search for a possible "solution," being able to try 10 billion Ns in one second. What would be your argument against such a project? Would you try some similar derivation to get an estimate of success? I know, brute forcing all integers is impossible, but you can even "imagine" a "superquantum (and now not existing) computer" which can do a lot of small integers in parallel. And yes, I know, probabilities aren't proofs, especially not for integers. One counterexample is enough: 3987^{12} + 4365^{12} = 4472^{12} Also interesting to see the accidental(?) order of magnitude of the numbers involved.

talaketu 10y ago

Wasn't the idea that he integrated on n for 100 to infinity, to arrive at that probability? So the probably should be interpreted as the probability of an eventual counter-example.