1. Basic number theory

I just started self learning basic number theory so please excuse me for any noob questions:

1. Prove that the fraction $\frac{n^3+2n}{n^4+3n^2+1}$ is in lowest terms for every positive integer $\mbox{n}$

2. Let $\{a,b,c\} \in \mathbb{N}$, show that $\frac{[a,b,c]^2}{[a,b][b,c][c,a]} = \frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}$

3. Prove that consecutive Fibonacci numbers are always relatively prime.

4. Show that $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$ can never be an integer. (I'm thinking of showing $\lim_{n \to \infty}\sum_{k=2}^{n}\frac{1}{k}$ converges to a value... or something along those lines, probs wrong heh)

Can anyone show me how to do these questions? Thank you.

2. First question

I would try a little rearrangement with the first question.

I would factor out n in the numerator into $n(n^2 + 2)$ and add and subtract 1 in the denominator which would transform the denominator into $n^4 + 3n^2 + 2 - 1$ which factors into $(n^2 +1)(n^2 + 2) - 1$. Now notice the common factor $(n^2 + 2)$ in both numerator and denominator.

I'll let you take it the rest of the way.

3. With question #4

As you may know, it's a harmonic series which sums to infinity as n increases without bound. You need to find a representative term for the sum of the first n terms and show that this term can never be an integer no matter what n is.

4. For the first one, you want to find polynomials $p(n), q(n)$ having integer coefficients such that

$p(n)(n^3+2n)+q(n)(n^4+3n^2+1)=1$.

For the third one, use the identity $f_n^2-f_{n-1}f_{n+1}=(-1)^{n+1}$.

For number 4, do not use wonderboy's advice, as you will not find a closed form for the $n$th harmonic number. Here is a hint : consider the highest power of $2$ less than $n$, say $2^m$. Then there is no other integer between $1$ and $n$ which is divisible by $2^m$. So what would you get if you put $1+\frac{1}{2}+...+\frac{1}{n}$ in a fraction having lowest terms, say $\frac{p}{q}$? (show that $p$ is odd, while $q$ is even!)

5. for the second one, hint: [m,n](m,n)=mn