# 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.