Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
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":

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

Date Subject Author
2/10/14 AP
2/10/14 Brian Q. Hutchings
2/10/14 Pubkeybreaker
2/10/14 Karl-Olav Nyberg
2/10/14 William Elliot
2/11/14 AP
2/11/14 Karl-Olav Nyberg
2/11/14 Robin Chapman
2/11/14 Karl-Olav Nyberg
2/11/14 Robin Chapman
2/11/14 Karl-Olav Nyberg
2/11/14 Karl-Olav Nyberg
2/15/14 Karl-Olav Nyberg
2/15/14 quasi
2/15/14 Karl-Olav Nyberg
2/15/14 quasi
2/11/14 William Elliot
2/11/14 Robin Chapman
2/11/14 William Elliot
2/13/14 quasi
2/14/14 Brian Q. Hutchings