3411. Euclid's theorem that there are infinitely many primes
You could also look up proofs for an infinity of primes of the form 4k+1, 4k-1, 6k+1 and so on
2. If a prime divides ab, then either p divides a or p divides b
Proof: If p does not divide a, then gcd (p,a) = 1
Hence there exist integers x and y such that xp + ya = 1
(for this look up Euclid's algorithm for finding gcd)
Multiply this eqn by b to get xpb + yab = b
Since xpb and yab are both divisible by p, LHS is divisible by p and hence so should RHS. This implies that p divides b. (this is written as p|b in textbooks)
3. The above theorem is the basis for the most common proofs for the Fundamental Theorem of Arithmetic - Unique Prime Factorisation of integers
If you are further interested there are other notable theorems"
1. Chebyshev's theorem or Bertrand's Postulate that there is atleast one prime p between n and 2n for n>1
2. Prime Number Theorem regarding density of primes
etc.