Proof by Absurdity: Infinity of Prime Numbers

Extract

[...] Let a be an integer and z be a prime integer, we then have then : z being a prime number, it belongs to the list and so z divides . As z divides using question we conclude that z divides x - . x - = therefore z divides 1 Or the only number dividing 1 is itself and 1 is not prime. z is therefore not prime. We arrive at a contradiction, and the initial hypothesis is therefore false. There is then an infinity of prime numbers. [...]

[...] But as we have : x = 2*d+1 here d = 3*5*7*11 x = 3*e+1 here e = 2*5*7*11 x = 5*f here f = 2*3*7*11 x = 7*g + 1 x = 11*h+1 We can assert that x is not divisible by that is to say x is not divisible by any of the prime numbers (our hypothesis being that there are only 5 prime numbers). We therefore arrive at a contradiction and this proves that our initial hypothesis is false. This is the principle of a proof by absurdity. is the largest prime number and x is greater than x is therefore not a prime number. x is then divisible by a prime number. [...]

[...] Is there an infinity of prime numbers? A. If q divides then there exists an integer c such that : For example q = 2 and a = 10, we have 2 divided by 10 And there is a c such that 10 = 2 * here c = 5 because 10 = 2 * 5 If q divides then there exists an integer such that For example b = 30, q remains unchanged. We have 10 divided by 30 and therefore there exists a such that 30 = 2 * here = 15 Then we replace a and b with their values, namely q*c for a and for b in the expression a - and then we factor by q. [...]