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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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Posts: 12,067
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 ...


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":


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

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


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