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let's do some fo fanna here if possible we are going to fight integers ABC so that a squared plus b squared equal to 4 c plus 3 and as always please pause the video and try this first okay I'll teach you guys to this well well let me show you guys my solution right here and of course you should share your station with us as well just leave a comment down below by the way I will show you guys my observations that making along the way first of all we do have a sum of two squares on the left-hand side so that means this is always going to be greater than equal to 0 right so what's that means that C has to be bigger than 0 as well because imagine if C is equal to negative 1 then you get 4 times negative 1 which is negative 4 plus 3 is negative 1 and your gear contradiction right because it's not possible for you to have a squared plus B squared is equal to negative 1 when a and B are just integers right so it's not possible but the truth is that it doesn't really matter if a PCR positive or negative all right hmm so that's kind of deep a lot of the consideration but anyway let's take a look of the following notice that 4 times T this right here it's always even and for you ask you to that the right hand side it's always art sometimes you have to be aware of the event and oddness okay the even is in at the honest and yes a little proof by the Simpsons he claimed that he found the context and poetry for most nasty room right by can easily disproved his counter example by using the even darkness anyway here is what we are going to observe I will just write down ok with no 40 plus 3 it's odd and let me just write down all the steps if you want to write on the solution this is what you will do I'll just say since we notice that 40 plus 3 is always odd so the rest is always odd what does that mean well open you have once in class the other being an odd number you must have one of them being even and the other one being art right because even plus art its art right well these are just squares of a and B if you have an even number you square that you can even if you an are number square you get out okay and if you have a square being even it has to be even and for example if a square is odd then that means a is art okay so they go like back-to-back but anyway we know that the right-hand side art that means either this what that it's art and the other is even okay that must be true and once again we're just going to break down what you said so we're just going to say that we must have one of the A or B to be even and once again I'm just saying amb okay because once again when you have a square it's even that means it's even or you have a tip and that means a square C but so they go back to back and we'll just want to talk about am like this but anyway I'm saying that either a what B has to be eaten and the other one has to be odd okay so that's the idea and now this is how we can do and this is one of the appreciation that if you guys have seen in the past this is the the coolest logarithm not just a joke I'll just say without loss of generality okay W lock this 10-spoke will tell loss of generality and what this is doing is that well you don't need to worry about you pick a to be eaten or beat with people because he's either A or B right so it doesn't really matter so just say without loss of generality let's say a let's just say a is even and B yes ah okay and seriously you can just change the am P now you can say a is art and a BC but that doesn't remain matter but you could put this down so that you can be such it okay so now you know a is even and B is art based on how we are saying right we may write when we have an even number we can say a it's equal to two times some other integer I'll just put on K for that and because b is art we can say B is equal to two times another integer it doesn't have to be the same as K so we just put them L but at the end you put down Plus what okay so this is the fourth ward even number and this is the form for the odd number and I was just say for L and K be in the integers this is like a horrible Z sorry tiny way yeah just like this okay so a Ln K are integers now we have an expression for a and B this is working - of course we're just plugging this back into the original equation so we know that a squared plus B squared now this would be okay a is 2k not like the 2k 18 for the basketball games but your 2k and that B is 2l plus 1 so just a plus 2 L plus 1 okay and now we can just do some algebra to K square is of course just for K Square and then for you multiply this out you get 4 L square plus 4 L plus 1 and here's the part that you have to kind of just pay attention to well notice that the first three terms that we have right here they all have a four times right so we can factor out the 4 so let's do that this is for parentheses and we have K squared plus l squared and then plus L like this and then we have the plus one after that and now take a look K and L are integer so right here this is must be your integer right so when you have a squared plus B squared notice the four has to be four times integer and then plus one but on the right hand side this is saying a square plus B Square the four has to be 4c plus three right four times integer plus three and this is the contradiction so what we can do is I can just tell you this right here which is never possible to be 4c plus three right because once again you can also use a modulo arithmetic if you want okay so you can't do that you can say the following right here another way to write it you can say that a squared plus B squared this right here it's congruent to 1 mod 4 because this part is the multiple of 4 and then when you divide this by 4 you get a remainder of 1 right so you can say this is congruent to 1 mod 4 but well you have 4c plus 3 this right here it's of course congruent to 3 when you mad for right so that's why I said this right here it's never possible to be the same as for 3 plus 3 right when you're talking about just integer solutions so the height here of all this is that this is actually not possible how does the here you guys that first let me just write down we conclude that the equation a squared plus B squared is equal to 4 C plus 3 has now inter technical these intercostal champa you know I don't really understand why people say InterPro because integral means integral but you don't know what me I would just say no integers students all right yeah so this is how you can do like an argument oops I put a box the wrong way but yeah I should put the part but yeah hopefully you guys will kill with that by the way I just only shakers I guess died about doing some numbers we prove what these proof right mmm yeah this is one of the things I really like when I was in school but anyway hopefully gets all like this a you guys are new to my channel please subscribe I like to make lots of math videos for yes and as always that's it okay before we go here's the bonus part is this true or false this is false it seems like to be a context and poor to D for most lastly room right but no no no no not just though why if you look at the left-hand side here this right here well you know it's an even number raised to a power this right here it's even right and you see that this number of course B&C is 1 so this is odd and when you raise that to a power this right here is odd but I'll need right hand side this is an even number when you raise that to a power you end up with IVA and what we're saying is that well you have an even plus an odd number this is equal to e but of course this right here is false right so keep this in mind this is the pretty good strategy to try to show equation has integer solution or not [Music] you you