Integer-valued function

  • Thread starter ypatia
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  • #1
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Main Question or Discussion Point

Is there any function (if any) f: Z -> Z such that
f(f(n))=-n , for every n belongs to Z(integers) ??


I think that there is not any function like the one described above but how can we prove it. Any ideas??
Thanks in Advance
 

Answers and Replies

  • #2
mathman
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How about f(n)=in?
 
  • #3
CRGreathouse
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Not integer-valued. (I assume if the OP meant Gaussian integers that would have been mentioned, since that's the obvious solution.)

I've been thinking about this for a few hours now and I can't see any way to do it, but I can't prove that it's impossible.
 
  • #4
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How about

for n>0
f(2n-1)=2n
f(2n)=-2n+1
f(-2n+1)=-2n
f(-2n)=2n-1

f(0)=0
 
  • #5
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Nice, chronon. Nice.
 
  • #6
matt grime
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Indeed - it's nice to visualize f as a piecewise permutation

(0)(-2,-1,2,1)(-4,-3,4,3)...(-2n,-2n+1,2n,2n-1)....

and recall that (abcd)^2=(ac)(bd)
 

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