Thread: Basic number theory

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

1. Prove that the fraction is in lowest terms for every positive integer

2. Let , show that

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

4. Show that can never be an integer. (I'm thinking of showing converges to a value... or something along those lines, probs wrong heh)

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

I would try a little rearrangement with the first question.

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

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

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.

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

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

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

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

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