# NT8

Prove that there are infinitely many primes of the form .

Suppose on the contrary that there are just a finite number of these primes, say *p*_{1}, ..., *p*_{n}. Let *a* = *p*_{1}...*p*_{n}. Obviously, *a* = (–1)^{n} modulo 6. If *n* is even, let *b* = *a* + 4 and otherwise let *b* = *a* + 6. Then *b* = 5 modulo 6 = –1 modulo 6. Moreover, any prime number *q* dividing both *a* and *b* will also divide either 4 or 6, hence such a *q* will equal 2 or 3. In particular, none of the *p*_{j} will divide *b*. Now write *b* as a product of prime numbers. Since *b* is odd, every prime factor *q* of *b* can be written as *q* = 6*r* + 1, *q* = 6*r* + 3 or *q* = 6*r* + 5. However, the last type (which is –1 modulo 6) is already ruled out, since none of the *p*_{j} divides *b*. The second type only permits *q* = 3. Therefore *b*, being the product of all its prime factors, is either 1 or 3 modulo 6 (since 3^{m} = 3 modulo 6), a contradiction which completes the proof.