Involutions |

Examples:We wish to prove that the number of divisors ofn Îis odd if and only ifN^{*}nis a square. In order to do this we consider the setM:= {d Îof these divisors and defineN| d divides n }This mapping is an involution, ift:M -> M :d -> n/d.n>1, and obviously| M |is odd if and only ifthas a fixed point, i.e. if and only if there exists a divisordsuch thatd=n/d, or, in other words, if and only ifn=d.^{2}A less trivial example is the following proof (due to D. Zagier) of the fact that every prime number which is congruent 1 modulo 4 can be expressed as a sum of two squares of positive natural numbers. Consider the set

The following map is an involution onS:= {(x,y,z) Î(N^{*})^{3}| x^{2}+4yz=p }.S(exercise):t:(x,y,z) -> (x+2z,z,y-x-z) if x<y-z,t:(x,y,z) -> (2y-x,y,x-y+z) if y-z<x<2y,This involution has exactly one fixed point, namelyt:(x,y,z) -> (x-2y,x-y+z,y) if x>2y.(1,1,k), ifp=4k+1, therefore| S |must be odd, and consequently the involutionpossesses a fixed point, too, which shows thats:(x,y,z) -> (x,z,y)p=x, a sum of two squares.^{2}+4y^{2}

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Involutions |