# Basic number theory

• December 21st 2009, 08:32 AM
usagi_killer
Basic number theory
I just started self learning basic number theory so please excuse me for any noob questions:

1. Prove that the fraction http://stuff.daniel15.com/cgi-bin/ma...E4+3n%5E2+1%7D is in lowest terms for every positive integer http://stuff.daniel15.com/cgi-bin/ma...%5Cmbox%7Bn%7D

2. Let http://stuff.daniel15.com/cgi-bin/ma...Cmathbb%7BN%7D, show that http://stuff.daniel15.com/cgi-bin/ma...29%28c,a%29%7D

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

4. Show that http://stuff.daniel15.com/cgi-bin/ma...%7B1%7D%7Bn%7D can never be an integer. (I'm thinking of showing http://stuff.daniel15.com/cgi-bin/ma...%7B1%7D%7Bk%7D converges to a value... or something along those lines, probs wrong heh)

Can anyone show me how to do these questions? Thank you.
• December 21st 2009, 01:36 PM
wonderboy1953
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.
• December 21st 2009, 01:42 PM
wonderboy1953
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.
• December 21st 2009, 02:45 PM
Bruno J.
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!)
• December 21st 2009, 04:37 PM
Shanks
for the second one, hint: [m,n](m,n)=mn