1. (Schumer 2.3 #1) Show that if n is a perfect square and a perfect cube, then it is a perfect sixth power.
2. (Schumer 2.3 # 3 (b)) Prove that there are infinitely many primes of the form 6k+5.
3. (Schumer 2.3 #8) If p and q are twin primes with 3 < p < q, then show that 9 divides pq+1 and that pq+1 is a perfect square. Recall that p, q are twin primes if q-p=2.
4. (Schumer 2.3 #14) Show that 3 is the only prime of the form k^4+k^2+1.
5. (Schumer 2.3 #15) We say that the real numbers r_1, ..., r_n are linearly independent [over the integers] if the only solution to k_1*r_1+k_2*r_2+...+k_n*r_n=0 where k_i are integers is k_1=k_2=...=k_n=0. Show that if p_1, p_2, ..., p_n are n distinct primes, then the real numbers log(p_1), log(p_2), ..., log(p_n) are linearly independent.