Heptadecagon and Fermat Primes (the math bit) - Numberphile

a very surprising thing is that gauss didn't actually do what i just did as far as i know or even try to or even care what he did was an abstract proof about the theory of equations he wanted he really wanted to prove something about roots of unity when they're called and he did in a very beautiful way before i explain that though let me tell you the way a mathematician would think about this construction here's a circle and now if i want to construct a 17 gun what i really want to do is to find 1 17 of the circle let's say it's here i don't know where it really is how what does it mean to construct that well if i construct that point then i could drop a perpendicular to the line so i really constructed this distance and if you think about it if this angle is a 17th of 360 degrees so this angle is 360 degrees over 17. mathematicians usually write that as 2 pi over 17 radians and so i'm really constructing a number which is the cosine of 2 pi over 17. or if you like i'm constructing a number which is the sine of 2 pi over 17 or if i think in complex numbers and i think that this is the imaginary axis here's i then i'm constructing a number which is cosine of 2 pi over 17 plus i times the sine of 2 pi over 17 and that's e to the 2 pi i over 17. that's a variant of euler's formula euler the one everybody knows is that if i took half a circle so 2 pi over 2 which is this is the angle 2 pi over 2 which of course is just pi then euler's formula says that if you take e to the pi i that's equal to minus one and there's minus one on the real axis so that's the usual way to write euler's formula but actually it works for any angle and so i'm trying to construct one of these numbers it's all they're all equivalent and so gauss was interested in what numbers you could construct with ruler and compass what what possible numbers and he proved a truly amazing theorem or we think he proved it uh i really do think he proved it but he didn't write down a full proof that was left for another 40 years he didn't bother again he was he believed in proving just a few really important things he said pauka said mathura is was his motto few but ripe anyway he exactly showed which numbers you could construct with ruler and compass and it turns out that the ones you can construct are the ones you could write in terms of square roots of things so what gauss actually did was to find a way of solving an equation like z to the 17 minus 1 equals 0. and when i say solving i mean writing down the roots in terms of of simpler things in this case in terms of square roots and that's what you need to be constructible so he showed this is going to be a big formula that the cosine of 2 pi over 17 was equal to minus 1 16 plus 1 16 times the square root of 17 plus 1 16 times the square root of 34 minus inside the square root sign 2 times the square root of 17. you're allowed to take iterated square roots in this plus 1 8 times a big square root of 17 plus 3 times the square root of 17 minus the square root of 34 minus 2 times the square root of 17 and then minus another 2 times the square root of 34 plus 2 times the square root of 17. you know what professor i think would have been easier for him to get the compass yeah well this this thing comes from an iterative iterative formula that he worked out he worked out a general method of solving certain kinds of equations in terms of simpler things like square roots and that's the method that led to the whole big breakthrough so this is a byproduct of his method and this construction that we saw which is so beautiful and complicated is a byproduct of this byproduct it was a sort of secondary thing for him but he was proud of it and he put it in the very end of his most famous book discus arithmetic as the sort of crowning chapter what has he done that showed the seventeen comm was adorable so if you can write the cosine of 2 pi over 17 in terms of square roots and adding and subtracting and multiplying and dividing that's exactly what you need to know to show that this segment the cosine of 2 pi over 17 can be constructed with ruler and compass once you've constructed that you make a perpendicular line and you look where it hits the circle and then you've got your 1 17th of a circle and you just measure it out and it goes around 17 times if you did it exactly right so that's that's what gauss knew would work and what does work at least in principle if not with a real ruler in compass was the 17 gone his aim was that his end game or was that like a byproduct i think it was a byproduct of his study of the roots of the equation z to the n minus one the gauss made a very deep study of these equations and what their roots look like in the complex numbers he used this to prove something much much more than just the 17 god so there are things called fermat primes pheromone numbers a pheromone number f sub k is 2 to the 2 to the k plus 1. now you've got to make it odd so it has to have a plus 1 have a chance of being prime and if you do this so what's f 0 f 0 2 well 2 to the 0 is 1. 2 to the 1 is two so two plus one is three that's a prime f one what's that that's two to the two to the one plus one two to the one is two two to the two is four 2 4 plus 1 is 5 so that's 5. another prime fairmon notice this by the way that's why they called pheromone numbers what's f2 2 to the 2 to the 2 plus 1. that well 2 to the 2 is 4 2 to the fourth is 16 for 16 plus 1 is 17. we've seen those guys before 3 5 were what the ancients knew 17 is what gauss did f3 i could go through this whole rigmarole again but let me just cut to the chase it's 257 and f that's a prime and f4 is 65 and 537 and that's another prime and fermat said look i've found a formula for primes now at the time of ferma nobody had the technique to figure out whether f5 was prime or not and so far my just guessed that the serial rules would continue and it took euler first to show that f5 is not prime and in fact if you look at f5 and you go on computing as long as we can compute with computers we've gotten to f32 none of these are prime so fermat was just well that's like the most that's the worst prime generating formula ever that's right it's terrible it drawn it generates exactly five prime numbers what gauss really proved i believe though he didn't write down the proof it was left to one cell 40 years later was that the regular n-gons that you can construct the n for which you can construct a regular n-gon are exactly the fermat primes themselves or products of distinct fairmont primes or any of those numbers times a power of 2 because you can divide sides in half and as long as you like so for instance you could in principle construct a regular 65 537 gun and in a trunk in gertingen there is a construction with ruler and compass of that end gun done by an amateur sometime between gauss's time and hours but i'm not sure that anyone has ever checked it to see whether it was right but you can construct in principle things like 257 times 17 times 64 gone but that's the only kinds of things you can construct so gauss really settled that question by using his technique for solving equations z to the n minus one in the back of his book this this famous famous book this question is arithmetic you can see my copy has taken some beating gauss is proud and he's so proud that he wrote down all the n-gons that you could construct in the first 300 numbers any all the numbers under 300 that is so you can see they're they're scattered around and there are a lot of them there are infinitely many so if you give me n i have to write n as a product of primes powers so i write n as a power of 2 times a power of some other prime k might be 0. so the 2 might not be there p1 to the a1 times p2 to the a2 etc where the pi are prime any prime any primes and then i look and i say if any ai is bigger than 1 then not possible that's 0.1 if any of the pi are not equal to f 0 f one f two or f three then not possible these are the firm uh those are the fairmont primes but only the first three that's all the fairmont primes we know well maybe there's another fairmont prime but any fairmont prime could go in that list people don't believe there are any more fairmont prime what's that f4 did i leave out f4 oh f4 yes or f or f4 or other fairmont primes then it wouldn't be possible so some of the you don't have to have all the fairmont primes but you can't have any other primes in the factorization so i guess the surprising thing is that there's a connection between how you factor a number into primes and whether you can write down a ruler and compass construction of a regular n-gun so so fermat thought he'd come up with a great prime generating formula yes right he hadn't but he'd indirectly come up with one of the components of constructing that's right that's right you can if you have two to the n plus 1 for other numbers you can show that n has to and if it's prime then it has to be a power of 2. so those are the only primes that you could write this kind of way right i would say that's the point and now i have my 17th of the circle finally by drawing this radius one radius and 2 radius so this distance from here to here is supposed to be the 17th of the circle now uh just for fun and you may have to go off camera for this brady let's see how how close i came all right now i would say i didn't get quite halfway between because i estimated that two three nine

Channel: Numberphile2

Views: 282,190

Keywords: Prime Number (Literature Subject), Mathematics (Field Of Study), Heptadecagon, Fermat Number

Id: oYlB5lUGlbw

Length: 12min 26sec (746 seconds)

Published: Mon Feb 16 2015