Homework 12

This assignment is due in class by 11am on Friday, December 5th. 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 nth 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 $c = 1105$.
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