quasi
Posts:
11,309
Registered:
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Re: on real part of [(1+isqrt(7))/2]^n
Posted:
Feb 13, 2014 11:46 PM


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 nonelementary 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

