You know Zagier’s brilliant-but-baffling “one sentence proof” that every prime of the form 4k+1 is the sum of two squares? It turns out there’s a lovely intuitive explanation of it! https://mathoverflow.net/a/299696/8217 pic.twitter.com/uoSLHwjbF5
U tweetove putem weba ili aplikacija drugih proizvođača možete dodati podatke o lokaciji, kao što su grad ili točna lokacija. Povijest lokacija tweetova uvijek možete izbrisati. Saznajte više
The idea, briefly, is: The transformation illustrated transforms every windmill to a different windmill of the same area; cross-shaped windmills (whose arms are the same width as the central square) are mapped to themselves.
There is at most one cross-shaped windmill with a given prime area, since the width of the arms must divide the total area (of a cross-shaped windmill). If p = 4k + 1 then there exists a cross-shaped windmill of area p (with four arms of width 1 and length k).
So… if p = 4k + 1 is prime, there are an *odd* number of windmills with area p.
Now consider the transformation that rotates the windmill arms by 90°. This doesn’t change the area either. Since there are an odd number of windmills of area p, the arm-rotation transformation must also have a fixed point, i.e. there must be a windmill with square arms.
So p is the sum of a square and four times a square – but four times a square is still a square!
The lattice theory proof is also a favorite of mine.
I don’t think I know that one. How does it go?
Twitter je možda preopterećen ili ima kratkotrajnih poteškoća u radu. Pokušajte ponovno ili potražite dodatne informacije u odjeljku Status Twittera.