Talk:Effective results in number theory
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[edit]"effectively computable" should be defined somehow ;-) --FvdP
While I revised the french translation of this article, I stumbled on a few sentences I could hardly understand:
- These included lower bounds for class numbers (ideal class groups for some families of number fields grow);: I understood & translated as : "lower bounds on how grows the ideal class group of some families of fields". I'll put that back in this article, but can someone confirm it's true ?
- Ineffective results are still being proved in the shape A or B, where we have no way of telling which. That's not clear (to me). Does this mean: no effective way to tell which of A or B, but an effective way to determine "A or B" ?
Also, about: taking much more care about proofs by contradiction: what kind of care ? Care of making them effective ?
--FvdP 20:47, 16 Mar 2004 (UTC)
Effectively computable - old-fashioned, see Church-Turing thesis. It's better to link to that, than explain it twice.
Growth of class numbers - the sense is OK, but the English expression above? Anyway, typically we want to bound from below (minoration) for h(-d), the class number of Q(√-d).
Example of a fairly recent ineffective result: one out of 3, 5, and 7 is a primitive root modulo p for infinitely many primes p; but we cannot compute which one, from the proof.
Charles Matthews 15:50, 17 Mar 2004 (UTC)