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Indeed this is correct, my choice was the first prime exceeding n but yours has the advantage of being constructive I guess :). For the record I never ended up solving c) when the restriction is in place. I don't know how difficult that part may be, but the rest is fairly straightforward.

Lucky no-one is asking you to then.

No worries, glad to help.

You can think of a mathematical argument as a chain of logic, a bunch of facts connected by logical steps. Eg/ x^2+2x+1=0 => (x+1)^2=0 => x+1=0 => x=-1. All we have shown here is that any solution to the original equation must be -1. This does NOT necessarily mean that -1 IS in fact a...

Actually, I apologise, what I meant to say was that x^2=a^2 does NOT imply that x=a. So you cant just square both sides of an equation you are trying to solve in general, you will add "fake" solutions.

(\sqrt{x})^2 \neq x.

There are multiple spellings.

http://en.wikipedia.org/wiki/Binomial_series

An algorithmic version of the sieve of eratosthenes :). PS The sum of the reciprocals of the primes up to n is asymptotic to log(log n) :), even better than log(n).

I meant lowering the exponent of n rather than the constant, mathematicians seldom care too much about the constant either. We also think about it as being 'about' blah by the way, we just need some kind of formal definition so this notion can be used in proofs etc.

Waking the cadaver- blood splattered satisfaction.

Haha but what if we bound the growth of something above by something of the order of n^2 but we don't know that this is the best possible bound? Some notation is needed here surely :). A mathematical version of "about" is the relation ~. We say f~g if lim f(x)/g(x)=1.

Things like big-O notation vary slightly depending on context...generally the source will give an explicit definition of the notation when they introduce it. Most commonly, if f:R->R is a function, we write O(f(x)) for an arbitrary function that is bounded above by a positive constant multiple...

"An unjustified axiom"? In what way can any axiom really be justified? It is simply a list of formal rules defining an object, from which we deduce these objects properties. The word non-rigourous is a bit vague for my liking, but in essense yeah we cannot set maths on as firm a foundation as we...

Have not studied this in too much depth before. I think an axiomatisation of arithmetic is just something that is a formal system roughly equivalent to the Peano axioms. Have not read much about mathematical idealism, what are its main features? Its often not as simple as just pigeonholing...

Rudin- Principles of mathematical analysis is an amazing book to accompany R&C analysis.

I agree. I think there are definite benefits of a thread like this (and a purpose distinct from that of the more specific threads in this subforum). Not the least of which is the fact that it is much less intimidating to ask a potentially 'stupid question' in discussions like this than it is to...

If you truly find it interesting than you will find it much MUCH easier than most to make progress. Getting to the level of being able to take second year courses is certainly doable if you as driven as you sound.

Its an example of why proving things with pictures can be problematic. There are lots of things like this, eg Gabriel's horn has infinite surface area but finite volume.

Yes although we lose uniqueness, things like i^i have infinitely many values....think of it as a much worse version of the problem we get when considering square roots of positive numbers, there are two! I explained how this works in the p00n thread, within the last two pages or so. PS. Also...