Question

If n is an integer such that (n - 7) * (n + 3) is a prime number, find all positive values of n other than 8. Pls don't spam or I might report you.

Answer 1

Answer:

There is no largest prime number.

To prove this, let’s consider only n prime numbers: p1, p2, …, pn. But no prime pi divides the number

N = p1 * p2 * … * pn + 1,

so N cannot be composite. This contradicts the fact that the set of primes is finite.

Exercise 1. Sequence an is defined recursively:



Prove that ai and aj, i ¹ j are relatively prime.

Hint: Prove that an+1 = a1a2…an + 1 and use Euclid’s theorem.

Exercise 2. Ferma numbers Fn (n ≥ 0) are positive integers of the form



Prove that Fi and Fj, i ≠ j are relatively prime.

Hint: Prove that Fn +1 = F0F1F2…Fn + 2 and use Euclid’s theorem.

Dirichlet’s theorem about arithmetic progressions:

For any two positive coprime integers a and b there are infinitely many primes of the form a + n*b, where n > 0.