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Topic: real parts of powers
Replies: 9   Last Post: Feb 15, 2014 1:41 PM

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Tucsondrew@me.com

Posts: 806
Registered: 5/24/13
Re: real parts of powers
Posted: Feb 14, 2014 5:35 PM
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On Friday, February 14, 2014 2:40:36 PM UTC-7, 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 non-real 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 non-real number.

> quasi

ZG

PS. If I'm completely missing something, please be polite.
Thank you.



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