HW 11, due Wed December 7

  • Use terms of the continued fraction expansion of \sqrt{7} to find a solution to Pell’s equation m^2 - 7 n^2 = 1.
  • Calculate the Bernoulli number B_{6}.
  • Check that the Bernoulli number formula for 1^3 + 2^3 + \cdots+ (n-1)^3 gives the usual formula (n (n-1)/2)^2.
  • Use the functional equation for the Riemann zeta function to calculate \zeta(-3).
  • If P1 and P2 are disjoint sets of primes and if both have a well-defined Dirichlet density, then show that densities satisfy d(P1+P2) = d(P1)+d(P2).
  • If P is a finite set of primes, show that it has Dirichlet density 0.
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