Theorems and proofs
Problem (6) is the famous twin-primes conjecture, and it is not known whether it is true or false. Of course the fact that there are infinitely many primes was known to Euclid, and his proof of it can be found in Chapter 3, section 5. It is also known that if a and r are integers whose greatest common divisor is 1, then there are infinitely many primes of the form a + A t , where k is a positive integer. This was proved by L. Dirichlet in 1837. A very particular case of this result is proved in exercises 3 to 7 of Chapter 3. Another related conjecture asks whether there are infinitely many primes p for which p + 2 and p + 6 are also prime; in this case too the answer is not known. However, there is only one prime p for which p + 2 and p + 4 are also prime; see exercise 9 of Chapter 3.