
Beal's Conjecture Revisited - dmit
http://norvig.com/beal.html
======
nkurz
_However, I made an error back in 2000: my program ruled out all cases where A
and B had a common factor, but actually, as long as C does not also share the
factor, it is a valid counterexample._

Is there a proof for Norvig's less stringent "misinterpretation" that only A
and B must share a prime factor? Or is it also unproved? What about C and just
one of the others? Is it clear that all these cases are true or false
together?

~~~
Grue3
I don't know how Peter Norvig missed that, but if A and B have a common prime
factor, then C obviously also shares that factor (by prime number theorem).
The same is true for any other pair.

~~~
JadeNB
As a tiny matter of terminology, "prime number theorem" is usually reserved
for the theorem on the asymptotic number of primes
([https://en.wikipedia.org/wiki/Prime_number_theorem](https://en.wikipedia.org/wiki/Prime_number_theorem)).
I would probably just say "by prime factorisation", or, if you want to be
really precise, "by Euclid's lemma"
([https://en.wikipedia.org/wiki/Euclid%27s_lemma](https://en.wikipedia.org/wiki/Euclid%27s_lemma)).

------
JadeNB
Aside from playing, what is the use of this? To be sure, there's nothing
_wrong_ with playing, but I think that intellectual honesty would demand that
Norvig acknowledge that this (running a straightforward Python program) is not
how the conjecture will eventually be resolved, even if it is false.

I suppose somebody has to check the small possibilities, but true numerical
search uses algorithms that are either incredibly optimised or run on huge
time scales in order to check truly huge numbers. See, for example, the
discussion of the Collatz conjecture at
[http://sweet.ua.pt/tos/3x+1.html](http://sweet.ua.pt/tos/3x+1.html) , where
experimental verification is currently up to 2^(60) \approx 10^(18).

~~~
dmit
To be fair, he does cover this in the conclusion at the end of this article
and mentions how everything started in the original post from 2000:

[http://norvig.com/beal2000.html](http://norvig.com/beal2000.html)

~~~
JadeNB
Good point! I just saw the beginning, had that negative reaction, and closed
the tab; I should have given Norvig more credit.

