
Re: real parts of powers
Posted:
Feb 14, 2014 5:35 PM


On Friday, February 14, 2014 2:40:36 PM UTC7, quasi wrote: > Warmup problem (not hard): > > Show that there exists a complex number c such that Re(c^n) > is nonzero for all positive integers n, and such that > > Re(c^n1) = Re(c^n2) = Re(c^n3) > > for some distinct positive integers n1,n2,n3. >
Are you asking for a nonreal complex number c?
If not, let c = 1. For any even positive integer n, we have (1)^n = 1. Similarly, for odd n, resulting in 1.
> Conjecture: > > There does not exist a complex number c such that Re(c^n) > > is nonzero for all positive integers n, and such that > > Re(c^n1) = Re(c^n2) = Re(c^n3) = Re(c^n4) > > where n1,n2,n3,n4 are distinct positive integers. >
Counterexample, 1, see above.
Unless, of course, you mean nonreal number.
> quasi
ZG
PS. If I'm completely missing something, please be polite. Thank you.

