Algebraic Proof That Two Is the First Prime

Theorem. It is algebraically proved that 2 is the first prime.

Proof. Suppose that p₁ is the first prime. Then p₂ = p₁ + 1 is the second prime. because no two consecutive numbers share the same prime factors., Suppose that c₁ = p₂ + 1 is another integer. Then it is possible to write c₁ = p₁². Collecting the results, one has p₁² =p₁ +2. This is a quadratic equation in p₁. The solution is {-1, 2}. We have proved that 2 is the prime. We also obtained an unexpected result that -1 is also a legitimate prime.