$d in mathbb{N}$, that is not a square, show that the continued fractions for following numbers are purely...
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Let $d in mathbb{N}$ such that $d$ not a square number. Now show
that the continued fractions for $sqrt{d} + lfloorsqrt{d}rfloor$
and $frac{1}{(sqrt{d} - lfloorsqrt{d}rfloor)}$ are purely
periodic.
Furthermore, show that there exists $m in N$ so that $sqrt{d}$ is
periodic from the second number, i.e. $[a_0$; $overline
> {a_1a_2,...,a_m}]$ .
I tried to solve this, but I do not have an idea where to start.
Some of my attempts:
$1)$ For irrational number a of a degree $2$, and assuming $a'$ is the other root of the minimal polynomial of $a$, that $-1 < a' < 0$.
$~~1.1)$ Define following series $x_n$ for $a =[a_0$; ${a_1,a_2,...}$], then, $x_{n+1}$ is the real number such that $a =[a_0$; ${a_1,a_2,..., x_{n+1}}$] and that $x_n$ = $a_n$ + $frac{1}{x_{n+1}}$.
$~~1.2)$ Show that for any $n inmathbb{N}$, $-1 < x_{n+1} < 0$ and that $a_n=lfloor$$frac{-1}{x_{n+1}}$$rfloor$.
I've shown $1.1$ and $1.2$
Thank you.
prime-numbers continued-fractions
edited Nov 21 at 12:10
Klangen
1,32811130
asked Aug 30 at 16:34
MicroT
235
add a comment |
up vote
2
down vote
favorite
Let $d in mathbb{N}$ such that $d$ not a square number. Now show
that the continued fractions for $sqrt{d} + lfloorsqrt{d}rfloor$
and $frac{1}{(sqrt{d} - lfloorsqrt{d}rfloor)}$ are purely
periodic.
Furthermore, show that there exists $m in N$ so that $sqrt{d}$ is
periodic from the second number, i.e. $[a_0$; $overline
> {a_1a_2,...,a_m}]$ .
I tried to solve this, but I do not have an idea where to start.
Some of my attempts:
$1)$ For irrational number a of a degree $2$, and assuming $a'$ is the other root of the minimal polynomial of $a$, that $-1 < a' < 0$.
$~~1.1)$ Define following series $x_n$ for $a =[a_0$; ${a_1,a_2,...}$], then, $x_{n+1}$ is the real number such that $a =[a_0$; ${a_1,a_2,..., x_{n+1}}$] and that $x_n$ = $a_n$ + $frac{1}{x_{n+1}}$.
$~~1.2)$ Show that for any $n inmathbb{N}$, $-1 < x_{n+1} < 0$ and that $a_n=lfloor$$frac{-1}{x_{n+1}}$$rfloor$.
I've shown $1.1$ and $1.2$
Thank you.
prime-numbers continued-fractions
edited Nov 21 at 12:10
Klangen
1,32811130
asked Aug 30 at 16:34
MicroT
235
Here, we ask that you show your efforts and thoughts. Do not merely ask us to do it for you.
– GEdgar
Aug 30 at 16:40
What book or notes are you using? Both your questions are standard results, this one has a pretty long answer, not really suitable for online
– Will Jagy
Aug 30 at 16:45
@WillJagy Hi, I'm using following book link. Where can I find said results?
– MicroT
Aug 31 at 15:08
@MicroT See the My research section of math.stackexchange.com/questions/2749487/…. It shows the 4 continued fraction references that I have found helpful. As suggested by Will Jagy's comment, it is better to find and explore good continued fraction references, than to attempt to derive results yourself.
– user2661923
Sep 1 at 17:29
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $d in mathbb{N}$ such that $d$ not a square number. Now show
that the continued fractions for $sqrt{d} + lfloorsqrt{d}rfloor$
and $frac{1}{(sqrt{d} - lfloorsqrt{d}rfloor)}$ are purely
periodic.
Furthermore, show that there exists $m in N$ so that $sqrt{d}$ is
periodic from the second number, i.e. $[a_0$; $overline
> {a_1a_2,...,a_m}]$ .
I tried to solve this, but I do not have an idea where to start.
Some of my attempts:
$1)$ For irrational number a of a degree $2$, and assuming $a'$ is the other root of the minimal polynomial of $a$, that $-1 < a' < 0$.
$~~1.1)$ Define following series $x_n$ for $a =[a_0$; ${a_1,a_2,...}$], then, $x_{n+1}$ is the real number such that $a =[a_0$; ${a_1,a_2,..., x_{n+1}}$] and that $x_n$ = $a_n$ + $frac{1}{x_{n+1}}$.
$~~1.2)$ Show that for any $n inmathbb{N}$, $-1 < x_{n+1} < 0$ and that $a_n=lfloor$$frac{-1}{x_{n+1}}$$rfloor$.
I've shown $1.1$ and $1.2$
Thank you.
prime-numbers continued-fractions
edited Nov 21 at 12:10
Klangen
1,32811130
asked Aug 30 at 16:34
MicroT
235
Let $d in mathbb{N}$ such that $d$ not a square number. Now show
that the continued fractions for $sqrt{d} + lfloorsqrt{d}rfloor$
and $frac{1}{(sqrt{d} - lfloorsqrt{d}rfloor)}$ are purely
periodic.
Furthermore, show that there exists $m in N$ so that $sqrt{d}$ is
periodic from the second number, i.e. $[a_0$; $overline
> {a_1a_2,...,a_m}]$ .
I tried to solve this, but I do not have an idea where to start.
Some of my attempts:
$1)$ For irrational number a of a degree $2$, and assuming $a'$ is the other root of the minimal polynomial of $a$, that $-1 < a' < 0$.
$~~1.1)$ Define following series $x_n$ for $a =[a_0$; ${a_1,a_2,...}$], then, $x_{n+1}$ is the real number such that $a =[a_0$; ${a_1,a_2,..., x_{n+1}}$] and that $x_n$ = $a_n$ + $frac{1}{x_{n+1}}$.
$~~1.2)$ Show that for any $n inmathbb{N}$, $-1 < x_{n+1} < 0$ and that $a_n=lfloor$$frac{-1}{x_{n+1}}$$rfloor$.
I've shown $1.1$ and $1.2$
Thank you.
prime-numbers continued-fractions
prime-numbers continued-fractions
edited Nov 21 at 12:10
Klangen
1,32811130
asked Aug 30 at 16:34
MicroT
235
edited Nov 21 at 12:10
Klangen
1,32811130
asked Aug 30 at 16:34
MicroT
235
edited Nov 21 at 12:10
Klangen
1,32811130
edited Nov 21 at 12:10
Klangen
1,32811130
edited Nov 21 at 12:10
Klangen
1,32811130
1,32811130
asked Aug 30 at 16:34
MicroT
235
asked Aug 30 at 16:34
MicroT
235
asked Aug 30 at 16:34
MicroT
235
235
Here, we ask that you show your efforts and thoughts. Do not merely ask us to do it for you.
– GEdgar
Aug 30 at 16:40
What book or notes are you using? Both your questions are standard results, this one has a pretty long answer, not really suitable for online
– Will Jagy
Aug 30 at 16:45
@WillJagy Hi, I'm using following book link. Where can I find said results?
– MicroT
Aug 31 at 15:08
@MicroT See the My research section of math.stackexchange.com/questions/2749487/…. It shows the 4 continued fraction references that I have found helpful. As suggested by Will Jagy's comment, it is better to find and explore good continued fraction references, than to attempt to derive results yourself.
– user2661923
Sep 1 at 17:29
add a comment |
Here, we ask that you show your efforts and thoughts. Do not merely ask us to do it for you.
– GEdgar
Aug 30 at 16:40
What book or notes are you using? Both your questions are standard results, this one has a pretty long answer, not really suitable for online
– Will Jagy
Aug 30 at 16:45
@WillJagy Hi, I'm using following book link. Where can I find said results?
– MicroT
Aug 31 at 15:08
@MicroT See the My research section of math.stackexchange.com/questions/2749487/…. It shows the 4 continued fraction references that I have found helpful. As suggested by Will Jagy's comment, it is better to find and explore good continued fraction references, than to attempt to derive results yourself.
– user2661923
Sep 1 at 17:29
Here, we ask that you show your efforts and thoughts. Do not merely ask us to do it for you.
– GEdgar
Aug 30 at 16:40
Here, we ask that you show your efforts and thoughts. Do not merely ask us to do it for you.
– GEdgar
Aug 30 at 16:40
What book or notes are you using? Both your questions are standard results, this one has a pretty long answer, not really suitable for online
– Will Jagy
Aug 30 at 16:45
What book or notes are you using? Both your questions are standard results, this one has a pretty long answer, not really suitable for online
– Will Jagy
Aug 30 at 16:45
@WillJagy Hi, I'm using following book link. Where can I find said results?
– MicroT
Aug 31 at 15:08
@WillJagy Hi, I'm using following book link. Where can I find said results?
– MicroT
Aug 31 at 15:08
@MicroT See the My research section of math.stackexchange.com/questions/2749487/…. It shows the 4 continued fraction references that I have found helpful. As suggested by Will Jagy's comment, it is better to find and explore good continued fraction references, than to attempt to derive results yourself.
– user2661923
Sep 1 at 17:29
@MicroT See the My research section of math.stackexchange.com/questions/2749487/…. It shows the 4 continued fraction references that I have found helpful. As suggested by Will Jagy's comment, it is better to find and explore good continued fraction references, than to attempt to derive results yourself.
– user2661923
Sep 1 at 17:29
add a comment |
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Here, we ask that you show your efforts and thoughts. Do not merely ask us to do it for you.
– GEdgar
Aug 30 at 16:40
What book or notes are you using? Both your questions are standard results, this one has a pretty long answer, not really suitable for online
– Will Jagy
Aug 30 at 16:45
@WillJagy Hi, I'm using following book link. Where can I find said results?
– MicroT
Aug 31 at 15:08
@MicroT See the My research section of math.stackexchange.com/questions/2749487/…. It shows the 4 continued fraction references that I have found helpful. As suggested by Will Jagy's comment, it is better to find and explore good continued fraction references, than to attempt to derive results yourself.
– user2661923
Sep 1 at 17:29