Brown Numbers - Numberphile

I know that from looking at number files how many of the viewers comment all the time about the brown paper that you use so I thought it'd be appropriate to introduce the brown numbers this is short but sweet and I think it could be a challenge for the uh mathematically in inclined number file viewers it's really straightforward Brown numbers are a pair of numbers a pair of integers okay which we let's call them m and n but they have a very special property their property is the following that n factorial + 1 must equal m^2 let me just remind you what a um n factorial is for example 5 factorial is 1 * 2 * 3 * 4 * 5 so n factorial is just doing that up to the integer n Okay so that's it that defines my brown number it's it's a pair of integers M and M such that n factorial + 1 is m^2 let me just show you a couple so five and four okay that's my M and N let's just let's check it uh so four factorial n is 4 + 1 well that's equal to 24 + 1 which is 25 which is 5^ s another one is 11 and let me give you another one Bray cuz you know so 5 factorial + 1 is equal to what's 5 factorial well if 4 factorial is 24 5 factorial is 24 * 5 which is 120 + 1 that's 121 and that's 11 squar okay so m is 11 so that's why that one works so let's show you another one um another pair is 71 and uh seven all right and you can go through the same thing if you work out 77 factorial plus one that turns out to be 71 squared I've shown you three yeah that's it that's all there is that's all there are that's it in fact um one of the greatest mathematicians of the 20th century uh Paul Eros um has actually conjectured that there are only three and that uh and that we that we won't get anymore and so the puzzle the challenge I thought was I mean it's so simple right just go and try and find some so if Paul Eros conjectures that there are no more yeah what does that mean he's just said what it's not proven no there's no proof of it that not that I'm aware of he must have some very good reason for conjecturing it and um I doubt that it was that he's literally gone through every single integer although though he worked as he worked that hard he could well have there are great stories about Eros quite a character who for for many many years he he was effectively homeless he didn't have a home and what he would do is he would show up at the his collaborator house knock with his bag of clothes knock on their door and and say when they answered it say is your mind open and that was it and they would collaborate with him on on mathematics and he would stay with them and then he'd move on to the next person I mean I could just make a conjecture right here and now and it could be called The Brady Haron conjecture yeah but that's not really allowed what credibility does a conjecture have to have before it's really a conjecture I think you have to probably demonstrate that it's it's held in lots and lots and lots of situations where it might fall down and that it's it's survived all of the possible obvious tests that you can make but of course we can't we can't test this up to infinite number of numbers so it would actually need a formal proof for it to work