Vanilla 1.1.9 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 20 of 20
how can I apply mathematical induction to prove k takes all the values m takes, having in mind that no one can prove that k does not take all values of m?
The form if this question is «How can I prove X, having in mind that no one can prove the negation of X?»
If I understand correctly, the answer is: no one knows...
You have the sentence, "If we prove k takes all values m takes then the Goldbach conjecture is proved." Then, you have the question "From the above facts, how can I apply mathematical induction to prove k takes all the values m takes?"
In other words, you are asking us how to apply mathematical induction to prove the Goldbach conjecture. This is not an acceptable question. What sort of answer are you expecting?
Ah well, if it should be true, then...
I can answer your question here. A large collection of examples is not a proof, unless it is large enough to be exhaustive. Here is a MathOverflow big-list question that contains lots of promising patterns with eventual counterexamples. None of those cases admit proofs by induction, even with zillions of affirmative examples, because the general claims are false. In the case under consideration, you cannot expect a proof by induction to appear without first revealing more fundamental structure in the primes. In particular, probabilistic arguments using the prime number theorem need to be supplemented with strong error bounds.
What you want to ask is «Can I use this idea to prove the Goldbach conjecture?».
This is not a good question for MO. This has already been stated clearly by Scott, who has even answered your question in the only possible way.
Maybe this thread should be closed?
Vassilis, Mariano and Scott are not alone in their opinions. While you may view the calculations you have given as logical facts or evidence, they do not amount to a proof; and moreover, asking people if there is a way to turn them into a proof is actually, from the point of view of mathematics, asking them to do all the work.
You say, originally
From the above we can show by shear calculation m takes all values k takes from 1 to 100 and from 100 to 1,000,000 and from 10^6 to 10^9 and because the relation of k to m is asymptotic, m will take all values k takes.
This is not proof, this is observation followed by an assertion ("m will take all values k takes").
My question is, is this form of mathematical induction correct or incorrect?
Incorrect.
If incorrect, what are the logical arguments to support such an opinion?
Do you mean, the logical arguments to support the opinion that this is incorrect? The burden of proof is not on us; the burden of proof is on you. What you have done may be, in your opinion, overwhelming evidence, but it is not a rigorous mathematical argument. As Scott has already said above, mathematical induction is not just a matter of observing many examples and saying "well, surely it must be true". If you dispute this basic premise, then I am sorry but it seems unlikely that there can be worthwhile engagement between you and this community.
If you post this question on MO, I will vote to close it. I also think that the present thread should be closed.
I agree with Yemon's last paragraph, and I can safely say that any question on possible proofs to Goldbach using the techniques outlined will be closed, and quickly. This is not an opinion, but borne out by experience with a) mathematics b) mathematicians and c) the rules of this site.
Let's leave all questions of substance aside. Could you please just accept that MO is not for presentation of half-way proofs and requests for feeback on them; in particular not; if they relate to famous conjectures, such as Goldbach. Please (re)read the FAQs. Thank you.
1 to 20 of 20