SELECTED CHAPTERS FROM ALGEBRA I. R. Shafarevich Preface

66 I. R. ShafarevichProblems1. Prove that there are innitely many primes of the form 3s +2.2. The same for the primes of the form 4s +3.3. Prove that each two numbers 2 2n +1 and 2 2m +1 are relatively prime.Deduce once more the innity ofthenumber of primes. [Hint. Assuming that p isa common divisor of two such numbers, nd the remainders of division of 2 2m and2 2n by p.]4. Let f(x) be a polynomial with integer coecients. Prove that there existinnitely many distinct prime divisors of its values f(1), f(2), . . . . (If you do notsucceed immediately, solve the problem for the polynomials of the rst and of thesecond degree.)5. Denote by p n the n-th prime in the natural order. Prove that p n+1