[FOM] Harvey's effective number theorists
Gabriel Stolzenberg
gstolzen at math.bu.edu
Wed Apr 12 15:12:49 EDT 2006
Harvey seems to be right about Roth's theorem and Falting's. I
was skeptical about both but now I think that, on the basis of the
evidence he's provided, it's not unreasonable to suppose that some
number theorists do, in each of these cases and no doubt many others,
express a strong interest in "effective" bounds and algorithms. For
an interesting test case, see Harold Stark's "Introduction to Number
Theory," (176-7, 179).
However, it remains to be checked whether number theorists mean
the same thing by these words that Harvey does. They are, after
all, classical mathematicians and "effective algorithm" is a term
used by classical mathematicians not constructivists.
Some Personal Comments
If, as Harvey suggests and his evidence seems to support, there is
an "intrinsic" interest in "effective" bounds and algorithms, then
I'm disappointed. I haven't followed these matters since the late
1970's but I confess that I had allowed myself to assume that number
theorists would have learned better by now.
Thinking back to that time, I wonder what Harvey would have made
of the case of a sign change for pi - li, with Littlewood's classical
proof and Skewes' number as the bound. For a time, it was the most
famous case of this kind. One can read about it in Ingham's "On the
Distribution of Prime Numbers" and, on the web, in Marcus du Sautoy's,
"The Music of the Primes" (127-131).
I also wonder what Harvey would have made of a brief exchange I
once had with Harold Stark about a first cousin to pi - li that he
had just proved. In the exchange, he played the role of Littlewood
and the question was whether we also needed a Skewes.
The Classical Proof
In Harvey's setup, there is always a classical proof. I would
like to know more about the role that having a classical proof plays
in making a number theorist want a constructive one and also about
the role that having no proof plays in making him seek a classical
one.
How Effective is "Effective"?
What are some good examples of a number theorist's "effective"
bounds and algorithms helping him to get realistic ones? Help that
he can't get just as easily from the classical proof.
Finally, I'd like to thank Harvey's first unnamed number theorist
for his comments and invite him to explain what makes the question of
getting an "effective" version of Falting's theorem that yields an
"effective" algorithm" for finding all rational points a "fundamental"
problem?
Gabriel
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