Homework 12

This assignment is due in class by 11am on Friday, December 5^{th}. Be sure to review the homework guidelines before getting started.

- Using Pari or
*Mathematica*if necessary, determine two values of*n*for which ibn-Qurra's theorem provides an amicable pair. Then find two values of*n*so that one ibn-Quarra's theorem does not provide an amicable pair. - Compute the aliquot sequence for 1264460. What does this tell you about 1264460?
- Find the odds that
*n*randomly chosen (positive) integers will be relatively prime. Your answer should be in terms of the Riemann Zeta function. Make sure to include a good explanation. - Let $F_n$ denote the n
^{th}Fibonacci number. Prove that for every positive integer*n*, we have $F_{2n-1} = F_n^2+F_{n-1}^2$ and $F_{2n} = F_{n+1}^2-F_{n-1}^2$. - Use the method of descent to prove that there are no integer solutions to the equation $a^2+b^2 = 3(c^2+d^2)$.
- Find the sibling triples for $c = 1105$.

page revision: 6, last edited: 02 Dec 2008 05:05