 Welcome back to another proof by contradiction example. This example is a little different than the first two because we're going to prove that something doesn't exist. Now how do you do that? This kind of situation where you're proving something fails to exist is a classic situation where proof by contradiction is useful. The theorem we're going to prove now says that there are no prime numbers greater than two that are congruent to 2 mod 4. Or if you wanted to say that differently if p is a prime number and greater than 2 then p is not congruent to 2 mod 4. So this will give us a chance to connect proof by contradiction back to a previous section where we learned about integer congruence. Generally speaking proof by contradiction is very useful when you're trying to prove a statement that is given in a negative form like something doesn't exist or two quantities are not equal which is what we saw in our first screencast or that an object does not belong to a certain set. To start a proof by contradiction here as always we're going to assume the negation of the statement we want to prove. Then we're going to do some valid mathematical steps and pull in some factual information to proceed from this assumption. The plan is that by doing this we'll eventually arrive at a contradiction and that will force us to reject the assumption thereby showing that the original statement is actually true. So the negation of this statement we want to prove is that there does exist a prime number bigger than two that is congruent to 2 mod 4. Let's call that prime number something say p. So at this point we know that p is a prime number bigger than two that is congruent to 2 mod 4. So let's work out what that means. By the definition of being congruent to 2 mod 4 it means that 4 divides the difference between p and 2. And then by the definition of divides this means that there's an integer k such that p minus 2 equals 4k. In other words p minus 2 is a multiple of 4. Now we haven't done a concept check in these videos for a while but I'd like to give you something kind of like that right now. I want you to pause the video and think about what comes next. What would you do next in the proof? Can you think of the correct next step in the reasoning here? And welcome back. You could do a lot of things here but maybe the most fruitful thing to do is to get p by itself. So if we add 2 to both sides of the equation p minus 2 equals 4k, we get of course p equals 4k plus 2. And here's where things start to unravel. You can clearly factor out a 2 from the terms on the right. And here is our contradiction. What we have now is a prime number that's been factored into two factors, a 2 and a 2k plus 1. Now note importantly that since p is known to be bigger than 2, we can conclude that k is not 0. That is the term in the parentheses here is not equal to 1. It's actually a positive integer bigger than 1. And here's where the contradiction happens of course. So on the one hand we said p was prime. But on the other hand here we are just factored it into two non-trivial factors. Now these two things can't be true at the same time. Either p cannot be so factored or it can. So there's our contradiction. And let's ask as we always do what got us to this contradiction in the first place. Well it was the assumption that there does actually exist a prime number bigger than 2 that's congruent to 2 mod 4. So the assumption that we made there causes a contradiction. And so it must be the case that those prime numbers cannot exist. So in other words, we've proven what we wanted. We've proven that no prime number can be congruent to 2 mod 4. So there again is a classic proof by contradiction used in a case when you have to prove that something does not exist. Thanks for watching.