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Topic: on real part of [(1+isqrt(7))/2]^n
Replies: 20   Last Post: Feb 15, 2014 5:52 AM

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quasi

Posts: 10,188
Registered: 7/15/05
Re: on real part of [(1+isqrt(7))/2]^n
Posted: Feb 13, 2014 11:46 PM
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AP wrote:
>
>Be a=(1+isqrt(7))/2 and a_n=the real part of a^n
>
>question : show lim |a_n | is +inf ?


This is not an easy problem.

Some years ago, I posted the following conjecture ...

Conjecture:

If c is an algebraic number with |c| > 1 such that Re(c^n) is
nonzero for all n in N, then |Re(c^n)| approaches infinity as
n approaches infinity.

The conjecture was resolved in the affirmative by "achille":

<https://groups.google.com/forum/#!msg/sci.math/XCljh7L4E0U/90q942YOjw4J>

The proof by achille used a non-elementary result, and at the
time, I felt that the claim "deserved" a more elementary proof.
I had some ideas for such a proof, but my attempts were
unsuccessful.

In any case, assuming achille's proof is correct, the answer
to your question is "yes".

quasi



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