Mixing
Demonstrate that these two Pell's equations have no integer solutions
$begingroup$
I would like to demonstrate that the following four Pell's equations have no integer solutions:
$x^2-82y^2=pm2$
$x^2-82y^2=pm3$
I do realise that such problems are often solved by algebraic manipulations, reducing modulo prime numbers, and arriving at some contradiction. After some blind trial and error fumbling in the dark with the above mentioned method, I have decided to consult the community.
All help or input would, as always, be highly appreciated.
number-theory elementary-number-theory pell-type-equations
asked Jan 31 at 22:40
Heinrich WagnerHeinrich Wagner
457211
$endgroup$
add a comment |
$begingroup$
I would like to demonstrate that the following four Pell's equations have no integer solutions:
$x^2-82y^2=pm2$
$x^2-82y^2=pm3$
I do realise that such problems are often solved by algebraic manipulations, reducing modulo prime numbers, and arriving at some contradiction. After some blind trial and error fumbling in the dark with the above mentioned method, I have decided to consult the community.
All help or input would, as always, be highly appreciated.
number-theory elementary-number-theory pell-type-equations
asked Jan 31 at 22:40
Heinrich WagnerHeinrich Wagner
457211
$endgroup$
add a comment |
$begingroup$
I would like to demonstrate that the following four Pell's equations have no integer solutions:
$x^2-82y^2=pm2$
$x^2-82y^2=pm3$
I do realise that such problems are often solved by algebraic manipulations, reducing modulo prime numbers, and arriving at some contradiction. After some blind trial and error fumbling in the dark with the above mentioned method, I have decided to consult the community.
All help or input would, as always, be highly appreciated.
number-theory elementary-number-theory pell-type-equations
asked Jan 31 at 22:40
Heinrich WagnerHeinrich Wagner
457211
$endgroup$
I would like to demonstrate that the following four Pell's equations have no integer solutions:
$x^2-82y^2=pm2$
$x^2-82y^2=pm3$
I do realise that such problems are often solved by algebraic manipulations, reducing modulo prime numbers, and arriving at some contradiction. After some blind trial and error fumbling in the dark with the above mentioned method, I have decided to consult the community.
All help or input would, as always, be highly appreciated.
number-theory elementary-number-theory pell-type-equations
number-theory elementary-number-theory pell-type-equations
asked Jan 31 at 22:40
Heinrich WagnerHeinrich Wagner
457211
asked Jan 31 at 22:40
Heinrich WagnerHeinrich Wagner
457211
asked Jan 31 at 22:40
Heinrich WagnerHeinrich Wagner
457211
asked Jan 31 at 22:40
Heinrich WagnerHeinrich Wagner
457211
asked Jan 31 at 22:40
Heinrich WagnerHeinrich Wagner
457211
457211
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
The second one is easy
$$x^2 - 82y^2 = pm3 Rightarrow x^2 equiv pm3 pmod{41}$$
but according to Euler's criterion
$$left(frac{pm3}{41}right) equiv left(pm3right)^{frac{41-1}{2}} pmod{41}$$
and $$left(pm3right)^{20} equiv -1 pmod{41}$$
as a result, there is no such $x$.
answered Jan 31 at 23:03
rtybasertybase
11.5k31534
$endgroup$
$begingroup$
There's also no solutions modulo $8$ to $x^2-82y^2=pm 3$.
$endgroup$
– Mike Bennett
Feb 1 at 16:41
add a comment |
$begingroup$
This is an example of a rather more general phenomenon. The continued fraction expansion of $sqrt{t^2+1}$ is just $[t,overline{2t}]$ and hence every convergent $p_i/q_i$ to $sqrt{t^2+1}$ has the property that
$$
p_i^2- (t^2+1) q_i^2 = pm 1.
$$
If we have $x^2-(t^2+1)y^2=k$ for a given integer $k$ and some integer $x$ with $y neq 0$, then
$$
left| sqrt{t^2+1} - frac{x}{y} right| = frac{|k|}{y^2 left|sqrt{t^2+1} + frac{x}{y} right|}
$$
and so $x/y$ is a convergent to $sqrt{t^2+1}$ provided, roughly, $|k| leq t$. It follows that the form $x^2-(t^2+1)y^2$ does not represent any non-square integers $k$ with $1< |k| < t$.
answered Feb 1 at 2:09
Mike BennettMike Bennett
2,41978
$endgroup$
$begingroup$
Good. I know few places that state clearly that primitively represented values (of $x^2 - n y^2$) up to $sqrt n$ occur as convergents for it. A slightly more general version is in L. E. Dickson, Introduction to Number Theory, about 1929. Theorem 85, attributed to Lagrange.
$endgroup$
– Will Jagy
Feb 1 at 2:27
add a comment |
$begingroup$
Recommend learning the following method for the continued fraction of $sqrt n.$
All small numbers $|k| < sqrt n$ that have a primitive representation as $x^2 - n y^2$ show up as $p^2 - n q^2,$ where $frac{p}{q}$ is a convergent for $sqrt n.$ As you can see, the only small represented numbers are $pm 1.$ See Theorem 5.1 in KCONRAD
Method described by Prof. Lubin at Continued fraction of $sqrt{67} - 4$
$$ sqrt { 82} = 9 + frac{ sqrt {82} - 9 }{ 1 } $$
$$ frac{ 1 }{ sqrt {82} - 9 } = frac{ sqrt {82} + 9 }{1 } = 18 + frac{ sqrt {82} - 9 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccc}
& & 9 & & 18 & & 18 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 9 }{ 1 } & & frac{ 163 }{ 18 } \
\
& 1 & & -1 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 82 cdot 0^2 = 1 & mbox{digit} & 9 \
frac{ 9 }{ 1 } & 9^2 - 82 cdot 1^2 = -1 & mbox{digit} & 18 \
frac{ 163 }{ 18 } & 163^2 - 82 cdot 18^2 = 1 & mbox{digit} & 18 \
end{array}
$$
========================================================
a different example:
$$ sqrt { 229} = 15 + frac{ sqrt {229} - 15 }{ 1 } $$
$$ frac{ 1 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{4 } = 7 + frac{ sqrt {229} - 13 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{15 } = 1 + frac{ sqrt {229} - 2 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 2 } = frac{ sqrt {229} + 2 }{15 } = 1 + frac{ sqrt {229} - 13 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{4 } = 7 + frac{ sqrt {229} - 15 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{1 } = 30 + frac{ sqrt {229} - 15 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccc}
& & 15 & & 7 & & 1 & & 1 & & 7 & & 30 & & 7 & & 1 & & 1 & & 7 & & 30 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 15 }{ 1 } & & frac{ 106 }{ 7 } & & frac{ 121 }{ 8 } & & frac{ 227 }{ 15 } & & frac{ 1710 }{ 113 } & & frac{ 51527 }{ 3405 } & & frac{ 362399 }{ 23948 } & & frac{ 413926 }{ 27353 } & & frac{ 776325 }{ 51301 } & & frac{ 5848201 }{ 386460 } \
\
& 1 & & -4 & & 15 & & -15 & & 4 & & -1 & & 4 & & -15 & & 15 & & -4 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 229 cdot 0^2 = 1 & mbox{digit} & 15 \
frac{ 15 }{ 1 } & 15^2 - 229 cdot 1^2 = -4 & mbox{digit} & 7 \
frac{ 106 }{ 7 } & 106^2 - 229 cdot 7^2 = 15 & mbox{digit} & 1 \
frac{ 121 }{ 8 } & 121^2 - 229 cdot 8^2 = -15 & mbox{digit} & 1 \
frac{ 227 }{ 15 } & 227^2 - 229 cdot 15^2 = 4 & mbox{digit} & 7 \
frac{ 1710 }{ 113 } & 1710^2 - 229 cdot 113^2 = -1 & mbox{digit} & 30 \
frac{ 51527 }{ 3405 } & 51527^2 - 229 cdot 3405^2 = 4 & mbox{digit} & 7 \
frac{ 362399 }{ 23948 } & 362399^2 - 229 cdot 23948^2 = -15 & mbox{digit} & 1 \
frac{ 413926 }{ 27353 } & 413926^2 - 229 cdot 27353^2 = 15 & mbox{digit} & 1 \
frac{ 776325 }{ 51301 } & 776325^2 - 229 cdot 51301^2 = -4 & mbox{digit} & 7 \
frac{ 5848201 }{ 386460 } & 5848201^2 - 229 cdot 386460^2 = 1 & mbox{digit} & 30 \
end{array}
$$
====================================
$$ sqrt { 106} = 10 + frac{ sqrt {106} - 10 }{ 1 } $$
$$ frac{ 1 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{6 } = 3 + frac{ sqrt {106} - 8 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{7 } = 2 + frac{ sqrt {106} - 6 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{10 } = 1 + frac{ sqrt {106} - 4 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{9 } = 1 + frac{ sqrt {106} - 5 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 5 } = frac{ sqrt {106} + 5 }{9 } = 1 + frac{ sqrt {106} - 4 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{10 } = 1 + frac{ sqrt {106} - 6 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{7 } = 2 + frac{ sqrt {106} - 8 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{6 } = 3 + frac{ sqrt {106} - 10 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{1 } = 20 + frac{ sqrt {106} - 10 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccccccccccccccccccc}
& & 10 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 10 }{ 1 } & & frac{ 31 }{ 3 } & & frac{ 72 }{ 7 } & & frac{ 103 }{ 10 } & & frac{ 175 }{ 17 } & & frac{ 278 }{ 27 } & & frac{ 453 }{ 44 } & & frac{ 1184 }{ 115 } & & frac{ 4005 }{ 389 } & & frac{ 81284 }{ 7895 } & & frac{ 247857 }{ 24074 } & & frac{ 576998 }{ 56043 } & & frac{ 824855 }{ 80117 } & & frac{ 1401853 }{ 136160 } & & frac{ 2226708 }{ 216277 } & & frac{ 3628561 }{ 352437 } & & frac{ 9483830 }{ 921151 } & & frac{ 32080051 }{ 3115890 } \
\
& 1 & & -6 & & 7 & & -10 & & 9 & & -9 & & 10 & & -7 & & 6 & & -1 & & 6 & & -7 & & 10 & & -9 & & 9 & & -10 & & 7 & & -6 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 106 cdot 0^2 = 1 & mbox{digit} & 10 \
frac{ 10 }{ 1 } & 10^2 - 106 cdot 1^2 = -6 & mbox{digit} & 3 \
frac{ 31 }{ 3 } & 31^2 - 106 cdot 3^2 = 7 & mbox{digit} & 2 \
frac{ 72 }{ 7 } & 72^2 - 106 cdot 7^2 = -10 & mbox{digit} & 1 \
frac{ 103 }{ 10 } & 103^2 - 106 cdot 10^2 = 9 & mbox{digit} & 1 \
frac{ 175 }{ 17 } & 175^2 - 106 cdot 17^2 = -9 & mbox{digit} & 1 \
frac{ 278 }{ 27 } & 278^2 - 106 cdot 27^2 = 10 & mbox{digit} & 1 \
frac{ 453 }{ 44 } & 453^2 - 106 cdot 44^2 = -7 & mbox{digit} & 2 \
frac{ 1184 }{ 115 } & 1184^2 - 106 cdot 115^2 = 6 & mbox{digit} & 3 \
frac{ 4005 }{ 389 } & 4005^2 - 106 cdot 389^2 = -1 & mbox{digit} & 20 \
frac{ 81284 }{ 7895 } & 81284^2 - 106 cdot 7895^2 = 6 & mbox{digit} & 3 \
frac{ 247857 }{ 24074 } & 247857^2 - 106 cdot 24074^2 = -7 & mbox{digit} & 2 \
frac{ 576998 }{ 56043 } & 576998^2 - 106 cdot 56043^2 = 10 & mbox{digit} & 1 \
frac{ 824855 }{ 80117 } & 824855^2 - 106 cdot 80117^2 = -9 & mbox{digit} & 1 \
frac{ 1401853 }{ 136160 } & 1401853^2 - 106 cdot 136160^2 = 9 & mbox{digit} & 1 \
frac{ 2226708 }{ 216277 } & 2226708^2 - 106 cdot 216277^2 = -10 & mbox{digit} & 1 \
frac{ 3628561 }{ 352437 } & 3628561^2 - 106 cdot 352437^2 = 7 & mbox{digit} & 2 \
frac{ 9483830 }{ 921151 } & 9483830^2 - 106 cdot 921151^2 = -6 & mbox{digit} & 3 \
frac{ 32080051 }{ 3115890 } & 32080051^2 - 106 cdot 3115890^2 = 1 & mbox{digit} & 20 \
end{array}
$$
answered Jan 31 at 23:17
Will JagyWill Jagy
104k5102201
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add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The second one is easy
$$x^2 - 82y^2 = pm3 Rightarrow x^2 equiv pm3 pmod{41}$$
but according to Euler's criterion
$$left(frac{pm3}{41}right) equiv left(pm3right)^{frac{41-1}{2}} pmod{41}$$
and $$left(pm3right)^{20} equiv -1 pmod{41}$$
as a result, there is no such $x$.
answered Jan 31 at 23:03
rtybasertybase
11.5k31534
$endgroup$
$begingroup$
There's also no solutions modulo $8$ to $x^2-82y^2=pm 3$.
$endgroup$
– Mike Bennett
Feb 1 at 16:41
add a comment |
$begingroup$
The second one is easy
$$x^2 - 82y^2 = pm3 Rightarrow x^2 equiv pm3 pmod{41}$$
but according to Euler's criterion
$$left(frac{pm3}{41}right) equiv left(pm3right)^{frac{41-1}{2}} pmod{41}$$
and $$left(pm3right)^{20} equiv -1 pmod{41}$$
as a result, there is no such $x$.
answered Jan 31 at 23:03
rtybasertybase
11.5k31534
$endgroup$
$begingroup$
There's also no solutions modulo $8$ to $x^2-82y^2=pm 3$.
$endgroup$
– Mike Bennett
Feb 1 at 16:41
add a comment |
$begingroup$
The second one is easy
$$x^2 - 82y^2 = pm3 Rightarrow x^2 equiv pm3 pmod{41}$$
but according to Euler's criterion
$$left(frac{pm3}{41}right) equiv left(pm3right)^{frac{41-1}{2}} pmod{41}$$
and $$left(pm3right)^{20} equiv -1 pmod{41}$$
as a result, there is no such $x$.
answered Jan 31 at 23:03
rtybasertybase
11.5k31534
$endgroup$
The second one is easy
$$x^2 - 82y^2 = pm3 Rightarrow x^2 equiv pm3 pmod{41}$$
but according to Euler's criterion
$$left(frac{pm3}{41}right) equiv left(pm3right)^{frac{41-1}{2}} pmod{41}$$
and $$left(pm3right)^{20} equiv -1 pmod{41}$$
as a result, there is no such $x$.
answered Jan 31 at 23:03
rtybasertybase
11.5k31534
answered Jan 31 at 23:03
rtybasertybase
11.5k31534
answered Jan 31 at 23:03
rtybasertybase
11.5k31534
answered Jan 31 at 23:03
rtybasertybase
11.5k31534
11.5k31534
$begingroup$
There's also no solutions modulo $8$ to $x^2-82y^2=pm 3$.
$endgroup$
– Mike Bennett
Feb 1 at 16:41
add a comment |
$begingroup$
There's also no solutions modulo $8$ to $x^2-82y^2=pm 3$.
$endgroup$
– Mike Bennett
Feb 1 at 16:41
$begingroup$
There's also no solutions modulo $8$ to $x^2-82y^2=pm 3$.
$endgroup$
– Mike Bennett
Feb 1 at 16:41
$begingroup$
There's also no solutions modulo $8$ to $x^2-82y^2=pm 3$.
$endgroup$
– Mike Bennett
Feb 1 at 16:41
add a comment |
$begingroup$
This is an example of a rather more general phenomenon. The continued fraction expansion of $sqrt{t^2+1}$ is just $[t,overline{2t}]$ and hence every convergent $p_i/q_i$ to $sqrt{t^2+1}$ has the property that
$$
p_i^2- (t^2+1) q_i^2 = pm 1.
$$
If we have $x^2-(t^2+1)y^2=k$ for a given integer $k$ and some integer $x$ with $y neq 0$, then
$$
left| sqrt{t^2+1} - frac{x}{y} right| = frac{|k|}{y^2 left|sqrt{t^2+1} + frac{x}{y} right|}
$$
and so $x/y$ is a convergent to $sqrt{t^2+1}$ provided, roughly, $|k| leq t$. It follows that the form $x^2-(t^2+1)y^2$ does not represent any non-square integers $k$ with $1< |k| < t$.
answered Feb 1 at 2:09
Mike BennettMike Bennett
2,41978
$endgroup$
$begingroup$
Good. I know few places that state clearly that primitively represented values (of $x^2 - n y^2$) up to $sqrt n$ occur as convergents for it. A slightly more general version is in L. E. Dickson, Introduction to Number Theory, about 1929. Theorem 85, attributed to Lagrange.
$endgroup$
– Will Jagy
Feb 1 at 2:27
add a comment |
$begingroup$
This is an example of a rather more general phenomenon. The continued fraction expansion of $sqrt{t^2+1}$ is just $[t,overline{2t}]$ and hence every convergent $p_i/q_i$ to $sqrt{t^2+1}$ has the property that
$$
p_i^2- (t^2+1) q_i^2 = pm 1.
$$
If we have $x^2-(t^2+1)y^2=k$ for a given integer $k$ and some integer $x$ with $y neq 0$, then
$$
left| sqrt{t^2+1} - frac{x}{y} right| = frac{|k|}{y^2 left|sqrt{t^2+1} + frac{x}{y} right|}
$$
and so $x/y$ is a convergent to $sqrt{t^2+1}$ provided, roughly, $|k| leq t$. It follows that the form $x^2-(t^2+1)y^2$ does not represent any non-square integers $k$ with $1< |k| < t$.
answered Feb 1 at 2:09
Mike BennettMike Bennett
2,41978
$endgroup$
$begingroup$
Good. I know few places that state clearly that primitively represented values (of $x^2 - n y^2$) up to $sqrt n$ occur as convergents for it. A slightly more general version is in L. E. Dickson, Introduction to Number Theory, about 1929. Theorem 85, attributed to Lagrange.
$endgroup$
– Will Jagy
Feb 1 at 2:27
add a comment |
$begingroup$
This is an example of a rather more general phenomenon. The continued fraction expansion of $sqrt{t^2+1}$ is just $[t,overline{2t}]$ and hence every convergent $p_i/q_i$ to $sqrt{t^2+1}$ has the property that
$$
p_i^2- (t^2+1) q_i^2 = pm 1.
$$
If we have $x^2-(t^2+1)y^2=k$ for a given integer $k$ and some integer $x$ with $y neq 0$, then
$$
left| sqrt{t^2+1} - frac{x}{y} right| = frac{|k|}{y^2 left|sqrt{t^2+1} + frac{x}{y} right|}
$$
and so $x/y$ is a convergent to $sqrt{t^2+1}$ provided, roughly, $|k| leq t$. It follows that the form $x^2-(t^2+1)y^2$ does not represent any non-square integers $k$ with $1< |k| < t$.
answered Feb 1 at 2:09
Mike BennettMike Bennett
2,41978
$endgroup$
This is an example of a rather more general phenomenon. The continued fraction expansion of $sqrt{t^2+1}$ is just $[t,overline{2t}]$ and hence every convergent $p_i/q_i$ to $sqrt{t^2+1}$ has the property that
$$
p_i^2- (t^2+1) q_i^2 = pm 1.
$$
If we have $x^2-(t^2+1)y^2=k$ for a given integer $k$ and some integer $x$ with $y neq 0$, then
$$
left| sqrt{t^2+1} - frac{x}{y} right| = frac{|k|}{y^2 left|sqrt{t^2+1} + frac{x}{y} right|}
$$
and so $x/y$ is a convergent to $sqrt{t^2+1}$ provided, roughly, $|k| leq t$. It follows that the form $x^2-(t^2+1)y^2$ does not represent any non-square integers $k$ with $1< |k| < t$.
answered Feb 1 at 2:09
Mike BennettMike Bennett
2,41978
answered Feb 1 at 2:09
Mike BennettMike Bennett
2,41978
answered Feb 1 at 2:09
Mike BennettMike Bennett
2,41978
answered Feb 1 at 2:09
Mike BennettMike Bennett
2,41978
2,41978
$begingroup$
Good. I know few places that state clearly that primitively represented values (of $x^2 - n y^2$) up to $sqrt n$ occur as convergents for it. A slightly more general version is in L. E. Dickson, Introduction to Number Theory, about 1929. Theorem 85, attributed to Lagrange.
$endgroup$
– Will Jagy
Feb 1 at 2:27
add a comment |
$begingroup$
Good. I know few places that state clearly that primitively represented values (of $x^2 - n y^2$) up to $sqrt n$ occur as convergents for it. A slightly more general version is in L. E. Dickson, Introduction to Number Theory, about 1929. Theorem 85, attributed to Lagrange.
$endgroup$
– Will Jagy
Feb 1 at 2:27
$begingroup$
Good. I know few places that state clearly that primitively represented values (of $x^2 - n y^2$) up to $sqrt n$ occur as convergents for it. A slightly more general version is in L. E. Dickson, Introduction to Number Theory, about 1929. Theorem 85, attributed to Lagrange.
$endgroup$
– Will Jagy
Feb 1 at 2:27
$begingroup$
Good. I know few places that state clearly that primitively represented values (of $x^2 - n y^2$) up to $sqrt n$ occur as convergents for it. A slightly more general version is in L. E. Dickson, Introduction to Number Theory, about 1929. Theorem 85, attributed to Lagrange.
$endgroup$
– Will Jagy
Feb 1 at 2:27
add a comment |
$begingroup$
Recommend learning the following method for the continued fraction of $sqrt n.$
All small numbers $|k| < sqrt n$ that have a primitive representation as $x^2 - n y^2$ show up as $p^2 - n q^2,$ where $frac{p}{q}$ is a convergent for $sqrt n.$ As you can see, the only small represented numbers are $pm 1.$ See Theorem 5.1 in KCONRAD
Method described by Prof. Lubin at Continued fraction of $sqrt{67} - 4$
$$ sqrt { 82} = 9 + frac{ sqrt {82} - 9 }{ 1 } $$
$$ frac{ 1 }{ sqrt {82} - 9 } = frac{ sqrt {82} + 9 }{1 } = 18 + frac{ sqrt {82} - 9 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccc}
& & 9 & & 18 & & 18 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 9 }{ 1 } & & frac{ 163 }{ 18 } \
\
& 1 & & -1 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 82 cdot 0^2 = 1 & mbox{digit} & 9 \
frac{ 9 }{ 1 } & 9^2 - 82 cdot 1^2 = -1 & mbox{digit} & 18 \
frac{ 163 }{ 18 } & 163^2 - 82 cdot 18^2 = 1 & mbox{digit} & 18 \
end{array}
$$
========================================================
a different example:
$$ sqrt { 229} = 15 + frac{ sqrt {229} - 15 }{ 1 } $$
$$ frac{ 1 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{4 } = 7 + frac{ sqrt {229} - 13 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{15 } = 1 + frac{ sqrt {229} - 2 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 2 } = frac{ sqrt {229} + 2 }{15 } = 1 + frac{ sqrt {229} - 13 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{4 } = 7 + frac{ sqrt {229} - 15 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{1 } = 30 + frac{ sqrt {229} - 15 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccc}
& & 15 & & 7 & & 1 & & 1 & & 7 & & 30 & & 7 & & 1 & & 1 & & 7 & & 30 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 15 }{ 1 } & & frac{ 106 }{ 7 } & & frac{ 121 }{ 8 } & & frac{ 227 }{ 15 } & & frac{ 1710 }{ 113 } & & frac{ 51527 }{ 3405 } & & frac{ 362399 }{ 23948 } & & frac{ 413926 }{ 27353 } & & frac{ 776325 }{ 51301 } & & frac{ 5848201 }{ 386460 } \
\
& 1 & & -4 & & 15 & & -15 & & 4 & & -1 & & 4 & & -15 & & 15 & & -4 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 229 cdot 0^2 = 1 & mbox{digit} & 15 \
frac{ 15 }{ 1 } & 15^2 - 229 cdot 1^2 = -4 & mbox{digit} & 7 \
frac{ 106 }{ 7 } & 106^2 - 229 cdot 7^2 = 15 & mbox{digit} & 1 \
frac{ 121 }{ 8 } & 121^2 - 229 cdot 8^2 = -15 & mbox{digit} & 1 \
frac{ 227 }{ 15 } & 227^2 - 229 cdot 15^2 = 4 & mbox{digit} & 7 \
frac{ 1710 }{ 113 } & 1710^2 - 229 cdot 113^2 = -1 & mbox{digit} & 30 \
frac{ 51527 }{ 3405 } & 51527^2 - 229 cdot 3405^2 = 4 & mbox{digit} & 7 \
frac{ 362399 }{ 23948 } & 362399^2 - 229 cdot 23948^2 = -15 & mbox{digit} & 1 \
frac{ 413926 }{ 27353 } & 413926^2 - 229 cdot 27353^2 = 15 & mbox{digit} & 1 \
frac{ 776325 }{ 51301 } & 776325^2 - 229 cdot 51301^2 = -4 & mbox{digit} & 7 \
frac{ 5848201 }{ 386460 } & 5848201^2 - 229 cdot 386460^2 = 1 & mbox{digit} & 30 \
end{array}
$$
====================================
$$ sqrt { 106} = 10 + frac{ sqrt {106} - 10 }{ 1 } $$
$$ frac{ 1 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{6 } = 3 + frac{ sqrt {106} - 8 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{7 } = 2 + frac{ sqrt {106} - 6 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{10 } = 1 + frac{ sqrt {106} - 4 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{9 } = 1 + frac{ sqrt {106} - 5 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 5 } = frac{ sqrt {106} + 5 }{9 } = 1 + frac{ sqrt {106} - 4 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{10 } = 1 + frac{ sqrt {106} - 6 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{7 } = 2 + frac{ sqrt {106} - 8 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{6 } = 3 + frac{ sqrt {106} - 10 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{1 } = 20 + frac{ sqrt {106} - 10 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccccccccccccccccccc}
& & 10 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 10 }{ 1 } & & frac{ 31 }{ 3 } & & frac{ 72 }{ 7 } & & frac{ 103 }{ 10 } & & frac{ 175 }{ 17 } & & frac{ 278 }{ 27 } & & frac{ 453 }{ 44 } & & frac{ 1184 }{ 115 } & & frac{ 4005 }{ 389 } & & frac{ 81284 }{ 7895 } & & frac{ 247857 }{ 24074 } & & frac{ 576998 }{ 56043 } & & frac{ 824855 }{ 80117 } & & frac{ 1401853 }{ 136160 } & & frac{ 2226708 }{ 216277 } & & frac{ 3628561 }{ 352437 } & & frac{ 9483830 }{ 921151 } & & frac{ 32080051 }{ 3115890 } \
\
& 1 & & -6 & & 7 & & -10 & & 9 & & -9 & & 10 & & -7 & & 6 & & -1 & & 6 & & -7 & & 10 & & -9 & & 9 & & -10 & & 7 & & -6 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 106 cdot 0^2 = 1 & mbox{digit} & 10 \
frac{ 10 }{ 1 } & 10^2 - 106 cdot 1^2 = -6 & mbox{digit} & 3 \
frac{ 31 }{ 3 } & 31^2 - 106 cdot 3^2 = 7 & mbox{digit} & 2 \
frac{ 72 }{ 7 } & 72^2 - 106 cdot 7^2 = -10 & mbox{digit} & 1 \
frac{ 103 }{ 10 } & 103^2 - 106 cdot 10^2 = 9 & mbox{digit} & 1 \
frac{ 175 }{ 17 } & 175^2 - 106 cdot 17^2 = -9 & mbox{digit} & 1 \
frac{ 278 }{ 27 } & 278^2 - 106 cdot 27^2 = 10 & mbox{digit} & 1 \
frac{ 453 }{ 44 } & 453^2 - 106 cdot 44^2 = -7 & mbox{digit} & 2 \
frac{ 1184 }{ 115 } & 1184^2 - 106 cdot 115^2 = 6 & mbox{digit} & 3 \
frac{ 4005 }{ 389 } & 4005^2 - 106 cdot 389^2 = -1 & mbox{digit} & 20 \
frac{ 81284 }{ 7895 } & 81284^2 - 106 cdot 7895^2 = 6 & mbox{digit} & 3 \
frac{ 247857 }{ 24074 } & 247857^2 - 106 cdot 24074^2 = -7 & mbox{digit} & 2 \
frac{ 576998 }{ 56043 } & 576998^2 - 106 cdot 56043^2 = 10 & mbox{digit} & 1 \
frac{ 824855 }{ 80117 } & 824855^2 - 106 cdot 80117^2 = -9 & mbox{digit} & 1 \
frac{ 1401853 }{ 136160 } & 1401853^2 - 106 cdot 136160^2 = 9 & mbox{digit} & 1 \
frac{ 2226708 }{ 216277 } & 2226708^2 - 106 cdot 216277^2 = -10 & mbox{digit} & 1 \
frac{ 3628561 }{ 352437 } & 3628561^2 - 106 cdot 352437^2 = 7 & mbox{digit} & 2 \
frac{ 9483830 }{ 921151 } & 9483830^2 - 106 cdot 921151^2 = -6 & mbox{digit} & 3 \
frac{ 32080051 }{ 3115890 } & 32080051^2 - 106 cdot 3115890^2 = 1 & mbox{digit} & 20 \
end{array}
$$
answered Jan 31 at 23:17
Will JagyWill Jagy
104k5102201
$endgroup$
add a comment |
$begingroup$
Recommend learning the following method for the continued fraction of $sqrt n.$
All small numbers $|k| < sqrt n$ that have a primitive representation as $x^2 - n y^2$ show up as $p^2 - n q^2,$ where $frac{p}{q}$ is a convergent for $sqrt n.$ As you can see, the only small represented numbers are $pm 1.$ See Theorem 5.1 in KCONRAD
Method described by Prof. Lubin at Continued fraction of $sqrt{67} - 4$
$$ sqrt { 82} = 9 + frac{ sqrt {82} - 9 }{ 1 } $$
$$ frac{ 1 }{ sqrt {82} - 9 } = frac{ sqrt {82} + 9 }{1 } = 18 + frac{ sqrt {82} - 9 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccc}
& & 9 & & 18 & & 18 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 9 }{ 1 } & & frac{ 163 }{ 18 } \
\
& 1 & & -1 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 82 cdot 0^2 = 1 & mbox{digit} & 9 \
frac{ 9 }{ 1 } & 9^2 - 82 cdot 1^2 = -1 & mbox{digit} & 18 \
frac{ 163 }{ 18 } & 163^2 - 82 cdot 18^2 = 1 & mbox{digit} & 18 \
end{array}
$$
========================================================
a different example:
$$ sqrt { 229} = 15 + frac{ sqrt {229} - 15 }{ 1 } $$
$$ frac{ 1 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{4 } = 7 + frac{ sqrt {229} - 13 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{15 } = 1 + frac{ sqrt {229} - 2 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 2 } = frac{ sqrt {229} + 2 }{15 } = 1 + frac{ sqrt {229} - 13 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{4 } = 7 + frac{ sqrt {229} - 15 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{1 } = 30 + frac{ sqrt {229} - 15 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccc}
& & 15 & & 7 & & 1 & & 1 & & 7 & & 30 & & 7 & & 1 & & 1 & & 7 & & 30 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 15 }{ 1 } & & frac{ 106 }{ 7 } & & frac{ 121 }{ 8 } & & frac{ 227 }{ 15 } & & frac{ 1710 }{ 113 } & & frac{ 51527 }{ 3405 } & & frac{ 362399 }{ 23948 } & & frac{ 413926 }{ 27353 } & & frac{ 776325 }{ 51301 } & & frac{ 5848201 }{ 386460 } \
\
& 1 & & -4 & & 15 & & -15 & & 4 & & -1 & & 4 & & -15 & & 15 & & -4 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 229 cdot 0^2 = 1 & mbox{digit} & 15 \
frac{ 15 }{ 1 } & 15^2 - 229 cdot 1^2 = -4 & mbox{digit} & 7 \
frac{ 106 }{ 7 } & 106^2 - 229 cdot 7^2 = 15 & mbox{digit} & 1 \
frac{ 121 }{ 8 } & 121^2 - 229 cdot 8^2 = -15 & mbox{digit} & 1 \
frac{ 227 }{ 15 } & 227^2 - 229 cdot 15^2 = 4 & mbox{digit} & 7 \
frac{ 1710 }{ 113 } & 1710^2 - 229 cdot 113^2 = -1 & mbox{digit} & 30 \
frac{ 51527 }{ 3405 } & 51527^2 - 229 cdot 3405^2 = 4 & mbox{digit} & 7 \
frac{ 362399 }{ 23948 } & 362399^2 - 229 cdot 23948^2 = -15 & mbox{digit} & 1 \
frac{ 413926 }{ 27353 } & 413926^2 - 229 cdot 27353^2 = 15 & mbox{digit} & 1 \
frac{ 776325 }{ 51301 } & 776325^2 - 229 cdot 51301^2 = -4 & mbox{digit} & 7 \
frac{ 5848201 }{ 386460 } & 5848201^2 - 229 cdot 386460^2 = 1 & mbox{digit} & 30 \
end{array}
$$
====================================
$$ sqrt { 106} = 10 + frac{ sqrt {106} - 10 }{ 1 } $$
$$ frac{ 1 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{6 } = 3 + frac{ sqrt {106} - 8 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{7 } = 2 + frac{ sqrt {106} - 6 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{10 } = 1 + frac{ sqrt {106} - 4 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{9 } = 1 + frac{ sqrt {106} - 5 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 5 } = frac{ sqrt {106} + 5 }{9 } = 1 + frac{ sqrt {106} - 4 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{10 } = 1 + frac{ sqrt {106} - 6 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{7 } = 2 + frac{ sqrt {106} - 8 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{6 } = 3 + frac{ sqrt {106} - 10 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{1 } = 20 + frac{ sqrt {106} - 10 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccccccccccccccccccc}
& & 10 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 10 }{ 1 } & & frac{ 31 }{ 3 } & & frac{ 72 }{ 7 } & & frac{ 103 }{ 10 } & & frac{ 175 }{ 17 } & & frac{ 278 }{ 27 } & & frac{ 453 }{ 44 } & & frac{ 1184 }{ 115 } & & frac{ 4005 }{ 389 } & & frac{ 81284 }{ 7895 } & & frac{ 247857 }{ 24074 } & & frac{ 576998 }{ 56043 } & & frac{ 824855 }{ 80117 } & & frac{ 1401853 }{ 136160 } & & frac{ 2226708 }{ 216277 } & & frac{ 3628561 }{ 352437 } & & frac{ 9483830 }{ 921151 } & & frac{ 32080051 }{ 3115890 } \
\
& 1 & & -6 & & 7 & & -10 & & 9 & & -9 & & 10 & & -7 & & 6 & & -1 & & 6 & & -7 & & 10 & & -9 & & 9 & & -10 & & 7 & & -6 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 106 cdot 0^2 = 1 & mbox{digit} & 10 \
frac{ 10 }{ 1 } & 10^2 - 106 cdot 1^2 = -6 & mbox{digit} & 3 \
frac{ 31 }{ 3 } & 31^2 - 106 cdot 3^2 = 7 & mbox{digit} & 2 \
frac{ 72 }{ 7 } & 72^2 - 106 cdot 7^2 = -10 & mbox{digit} & 1 \
frac{ 103 }{ 10 } & 103^2 - 106 cdot 10^2 = 9 & mbox{digit} & 1 \
frac{ 175 }{ 17 } & 175^2 - 106 cdot 17^2 = -9 & mbox{digit} & 1 \
frac{ 278 }{ 27 } & 278^2 - 106 cdot 27^2 = 10 & mbox{digit} & 1 \
frac{ 453 }{ 44 } & 453^2 - 106 cdot 44^2 = -7 & mbox{digit} & 2 \
frac{ 1184 }{ 115 } & 1184^2 - 106 cdot 115^2 = 6 & mbox{digit} & 3 \
frac{ 4005 }{ 389 } & 4005^2 - 106 cdot 389^2 = -1 & mbox{digit} & 20 \
frac{ 81284 }{ 7895 } & 81284^2 - 106 cdot 7895^2 = 6 & mbox{digit} & 3 \
frac{ 247857 }{ 24074 } & 247857^2 - 106 cdot 24074^2 = -7 & mbox{digit} & 2 \
frac{ 576998 }{ 56043 } & 576998^2 - 106 cdot 56043^2 = 10 & mbox{digit} & 1 \
frac{ 824855 }{ 80117 } & 824855^2 - 106 cdot 80117^2 = -9 & mbox{digit} & 1 \
frac{ 1401853 }{ 136160 } & 1401853^2 - 106 cdot 136160^2 = 9 & mbox{digit} & 1 \
frac{ 2226708 }{ 216277 } & 2226708^2 - 106 cdot 216277^2 = -10 & mbox{digit} & 1 \
frac{ 3628561 }{ 352437 } & 3628561^2 - 106 cdot 352437^2 = 7 & mbox{digit} & 2 \
frac{ 9483830 }{ 921151 } & 9483830^2 - 106 cdot 921151^2 = -6 & mbox{digit} & 3 \
frac{ 32080051 }{ 3115890 } & 32080051^2 - 106 cdot 3115890^2 = 1 & mbox{digit} & 20 \
end{array}
$$
answered Jan 31 at 23:17
Will JagyWill Jagy
104k5102201
$endgroup$
add a comment |
$begingroup$
Recommend learning the following method for the continued fraction of $sqrt n.$
All small numbers $|k| < sqrt n$ that have a primitive representation as $x^2 - n y^2$ show up as $p^2 - n q^2,$ where $frac{p}{q}$ is a convergent for $sqrt n.$ As you can see, the only small represented numbers are $pm 1.$ See Theorem 5.1 in KCONRAD
Method described by Prof. Lubin at Continued fraction of $sqrt{67} - 4$
$$ sqrt { 82} = 9 + frac{ sqrt {82} - 9 }{ 1 } $$
$$ frac{ 1 }{ sqrt {82} - 9 } = frac{ sqrt {82} + 9 }{1 } = 18 + frac{ sqrt {82} - 9 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccc}
& & 9 & & 18 & & 18 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 9 }{ 1 } & & frac{ 163 }{ 18 } \
\
& 1 & & -1 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 82 cdot 0^2 = 1 & mbox{digit} & 9 \
frac{ 9 }{ 1 } & 9^2 - 82 cdot 1^2 = -1 & mbox{digit} & 18 \
frac{ 163 }{ 18 } & 163^2 - 82 cdot 18^2 = 1 & mbox{digit} & 18 \
end{array}
$$
========================================================
a different example:
$$ sqrt { 229} = 15 + frac{ sqrt {229} - 15 }{ 1 } $$
$$ frac{ 1 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{4 } = 7 + frac{ sqrt {229} - 13 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{15 } = 1 + frac{ sqrt {229} - 2 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 2 } = frac{ sqrt {229} + 2 }{15 } = 1 + frac{ sqrt {229} - 13 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{4 } = 7 + frac{ sqrt {229} - 15 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{1 } = 30 + frac{ sqrt {229} - 15 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccc}
& & 15 & & 7 & & 1 & & 1 & & 7 & & 30 & & 7 & & 1 & & 1 & & 7 & & 30 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 15 }{ 1 } & & frac{ 106 }{ 7 } & & frac{ 121 }{ 8 } & & frac{ 227 }{ 15 } & & frac{ 1710 }{ 113 } & & frac{ 51527 }{ 3405 } & & frac{ 362399 }{ 23948 } & & frac{ 413926 }{ 27353 } & & frac{ 776325 }{ 51301 } & & frac{ 5848201 }{ 386460 } \
\
& 1 & & -4 & & 15 & & -15 & & 4 & & -1 & & 4 & & -15 & & 15 & & -4 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 229 cdot 0^2 = 1 & mbox{digit} & 15 \
frac{ 15 }{ 1 } & 15^2 - 229 cdot 1^2 = -4 & mbox{digit} & 7 \
frac{ 106 }{ 7 } & 106^2 - 229 cdot 7^2 = 15 & mbox{digit} & 1 \
frac{ 121 }{ 8 } & 121^2 - 229 cdot 8^2 = -15 & mbox{digit} & 1 \
frac{ 227 }{ 15 } & 227^2 - 229 cdot 15^2 = 4 & mbox{digit} & 7 \
frac{ 1710 }{ 113 } & 1710^2 - 229 cdot 113^2 = -1 & mbox{digit} & 30 \
frac{ 51527 }{ 3405 } & 51527^2 - 229 cdot 3405^2 = 4 & mbox{digit} & 7 \
frac{ 362399 }{ 23948 } & 362399^2 - 229 cdot 23948^2 = -15 & mbox{digit} & 1 \
frac{ 413926 }{ 27353 } & 413926^2 - 229 cdot 27353^2 = 15 & mbox{digit} & 1 \
frac{ 776325 }{ 51301 } & 776325^2 - 229 cdot 51301^2 = -4 & mbox{digit} & 7 \
frac{ 5848201 }{ 386460 } & 5848201^2 - 229 cdot 386460^2 = 1 & mbox{digit} & 30 \
end{array}
$$
====================================
$$ sqrt { 106} = 10 + frac{ sqrt {106} - 10 }{ 1 } $$
$$ frac{ 1 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{6 } = 3 + frac{ sqrt {106} - 8 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{7 } = 2 + frac{ sqrt {106} - 6 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{10 } = 1 + frac{ sqrt {106} - 4 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{9 } = 1 + frac{ sqrt {106} - 5 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 5 } = frac{ sqrt {106} + 5 }{9 } = 1 + frac{ sqrt {106} - 4 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{10 } = 1 + frac{ sqrt {106} - 6 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{7 } = 2 + frac{ sqrt {106} - 8 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{6 } = 3 + frac{ sqrt {106} - 10 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{1 } = 20 + frac{ sqrt {106} - 10 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccccccccccccccccccc}
& & 10 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 10 }{ 1 } & & frac{ 31 }{ 3 } & & frac{ 72 }{ 7 } & & frac{ 103 }{ 10 } & & frac{ 175 }{ 17 } & & frac{ 278 }{ 27 } & & frac{ 453 }{ 44 } & & frac{ 1184 }{ 115 } & & frac{ 4005 }{ 389 } & & frac{ 81284 }{ 7895 } & & frac{ 247857 }{ 24074 } & & frac{ 576998 }{ 56043 } & & frac{ 824855 }{ 80117 } & & frac{ 1401853 }{ 136160 } & & frac{ 2226708 }{ 216277 } & & frac{ 3628561 }{ 352437 } & & frac{ 9483830 }{ 921151 } & & frac{ 32080051 }{ 3115890 } \
\
& 1 & & -6 & & 7 & & -10 & & 9 & & -9 & & 10 & & -7 & & 6 & & -1 & & 6 & & -7 & & 10 & & -9 & & 9 & & -10 & & 7 & & -6 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 106 cdot 0^2 = 1 & mbox{digit} & 10 \
frac{ 10 }{ 1 } & 10^2 - 106 cdot 1^2 = -6 & mbox{digit} & 3 \
frac{ 31 }{ 3 } & 31^2 - 106 cdot 3^2 = 7 & mbox{digit} & 2 \
frac{ 72 }{ 7 } & 72^2 - 106 cdot 7^2 = -10 & mbox{digit} & 1 \
frac{ 103 }{ 10 } & 103^2 - 106 cdot 10^2 = 9 & mbox{digit} & 1 \
frac{ 175 }{ 17 } & 175^2 - 106 cdot 17^2 = -9 & mbox{digit} & 1 \
frac{ 278 }{ 27 } & 278^2 - 106 cdot 27^2 = 10 & mbox{digit} & 1 \
frac{ 453 }{ 44 } & 453^2 - 106 cdot 44^2 = -7 & mbox{digit} & 2 \
frac{ 1184 }{ 115 } & 1184^2 - 106 cdot 115^2 = 6 & mbox{digit} & 3 \
frac{ 4005 }{ 389 } & 4005^2 - 106 cdot 389^2 = -1 & mbox{digit} & 20 \
frac{ 81284 }{ 7895 } & 81284^2 - 106 cdot 7895^2 = 6 & mbox{digit} & 3 \
frac{ 247857 }{ 24074 } & 247857^2 - 106 cdot 24074^2 = -7 & mbox{digit} & 2 \
frac{ 576998 }{ 56043 } & 576998^2 - 106 cdot 56043^2 = 10 & mbox{digit} & 1 \
frac{ 824855 }{ 80117 } & 824855^2 - 106 cdot 80117^2 = -9 & mbox{digit} & 1 \
frac{ 1401853 }{ 136160 } & 1401853^2 - 106 cdot 136160^2 = 9 & mbox{digit} & 1 \
frac{ 2226708 }{ 216277 } & 2226708^2 - 106 cdot 216277^2 = -10 & mbox{digit} & 1 \
frac{ 3628561 }{ 352437 } & 3628561^2 - 106 cdot 352437^2 = 7 & mbox{digit} & 2 \
frac{ 9483830 }{ 921151 } & 9483830^2 - 106 cdot 921151^2 = -6 & mbox{digit} & 3 \
frac{ 32080051 }{ 3115890 } & 32080051^2 - 106 cdot 3115890^2 = 1 & mbox{digit} & 20 \
end{array}
$$
answered Jan 31 at 23:17
Will JagyWill Jagy
104k5102201
$endgroup$
Recommend learning the following method for the continued fraction of $sqrt n.$
All small numbers $|k| < sqrt n$ that have a primitive representation as $x^2 - n y^2$ show up as $p^2 - n q^2,$ where $frac{p}{q}$ is a convergent for $sqrt n.$ As you can see, the only small represented numbers are $pm 1.$ See Theorem 5.1 in KCONRAD
Method described by Prof. Lubin at Continued fraction of $sqrt{67} - 4$
$$ sqrt { 82} = 9 + frac{ sqrt {82} - 9 }{ 1 } $$
$$ frac{ 1 }{ sqrt {82} - 9 } = frac{ sqrt {82} + 9 }{1 } = 18 + frac{ sqrt {82} - 9 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccc}
& & 9 & & 18 & & 18 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 9 }{ 1 } & & frac{ 163 }{ 18 } \
\
& 1 & & -1 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 82 cdot 0^2 = 1 & mbox{digit} & 9 \
frac{ 9 }{ 1 } & 9^2 - 82 cdot 1^2 = -1 & mbox{digit} & 18 \
frac{ 163 }{ 18 } & 163^2 - 82 cdot 18^2 = 1 & mbox{digit} & 18 \
end{array}
$$
========================================================
a different example:
$$ sqrt { 229} = 15 + frac{ sqrt {229} - 15 }{ 1 } $$
$$ frac{ 1 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{4 } = 7 + frac{ sqrt {229} - 13 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{15 } = 1 + frac{ sqrt {229} - 2 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 2 } = frac{ sqrt {229} + 2 }{15 } = 1 + frac{ sqrt {229} - 13 }{15 } $$
$$ frac{ 15 }{ sqrt {229} - 13 } = frac{ sqrt {229} + 13 }{4 } = 7 + frac{ sqrt {229} - 15 }{4 } $$
$$ frac{ 4 }{ sqrt {229} - 15 } = frac{ sqrt {229} + 15 }{1 } = 30 + frac{ sqrt {229} - 15 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccc}
& & 15 & & 7 & & 1 & & 1 & & 7 & & 30 & & 7 & & 1 & & 1 & & 7 & & 30 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 15 }{ 1 } & & frac{ 106 }{ 7 } & & frac{ 121 }{ 8 } & & frac{ 227 }{ 15 } & & frac{ 1710 }{ 113 } & & frac{ 51527 }{ 3405 } & & frac{ 362399 }{ 23948 } & & frac{ 413926 }{ 27353 } & & frac{ 776325 }{ 51301 } & & frac{ 5848201 }{ 386460 } \
\
& 1 & & -4 & & 15 & & -15 & & 4 & & -1 & & 4 & & -15 & & 15 & & -4 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 229 cdot 0^2 = 1 & mbox{digit} & 15 \
frac{ 15 }{ 1 } & 15^2 - 229 cdot 1^2 = -4 & mbox{digit} & 7 \
frac{ 106 }{ 7 } & 106^2 - 229 cdot 7^2 = 15 & mbox{digit} & 1 \
frac{ 121 }{ 8 } & 121^2 - 229 cdot 8^2 = -15 & mbox{digit} & 1 \
frac{ 227 }{ 15 } & 227^2 - 229 cdot 15^2 = 4 & mbox{digit} & 7 \
frac{ 1710 }{ 113 } & 1710^2 - 229 cdot 113^2 = -1 & mbox{digit} & 30 \
frac{ 51527 }{ 3405 } & 51527^2 - 229 cdot 3405^2 = 4 & mbox{digit} & 7 \
frac{ 362399 }{ 23948 } & 362399^2 - 229 cdot 23948^2 = -15 & mbox{digit} & 1 \
frac{ 413926 }{ 27353 } & 413926^2 - 229 cdot 27353^2 = 15 & mbox{digit} & 1 \
frac{ 776325 }{ 51301 } & 776325^2 - 229 cdot 51301^2 = -4 & mbox{digit} & 7 \
frac{ 5848201 }{ 386460 } & 5848201^2 - 229 cdot 386460^2 = 1 & mbox{digit} & 30 \
end{array}
$$
====================================
$$ sqrt { 106} = 10 + frac{ sqrt {106} - 10 }{ 1 } $$
$$ frac{ 1 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{6 } = 3 + frac{ sqrt {106} - 8 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{7 } = 2 + frac{ sqrt {106} - 6 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{10 } = 1 + frac{ sqrt {106} - 4 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{9 } = 1 + frac{ sqrt {106} - 5 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 5 } = frac{ sqrt {106} + 5 }{9 } = 1 + frac{ sqrt {106} - 4 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{10 } = 1 + frac{ sqrt {106} - 6 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{7 } = 2 + frac{ sqrt {106} - 8 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{6 } = 3 + frac{ sqrt {106} - 10 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{1 } = 20 + frac{ sqrt {106} - 10 }{1 } $$
Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccccccccccccccccccc}
& & 10 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 10 }{ 1 } & & frac{ 31 }{ 3 } & & frac{ 72 }{ 7 } & & frac{ 103 }{ 10 } & & frac{ 175 }{ 17 } & & frac{ 278 }{ 27 } & & frac{ 453 }{ 44 } & & frac{ 1184 }{ 115 } & & frac{ 4005 }{ 389 } & & frac{ 81284 }{ 7895 } & & frac{ 247857 }{ 24074 } & & frac{ 576998 }{ 56043 } & & frac{ 824855 }{ 80117 } & & frac{ 1401853 }{ 136160 } & & frac{ 2226708 }{ 216277 } & & frac{ 3628561 }{ 352437 } & & frac{ 9483830 }{ 921151 } & & frac{ 32080051 }{ 3115890 } \
\
& 1 & & -6 & & 7 & & -10 & & 9 & & -9 & & 10 & & -7 & & 6 & & -1 & & 6 & & -7 & & 10 & & -9 & & 9 & & -10 & & 7 & & -6 & & 1
end{array}
$$
$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 106 cdot 0^2 = 1 & mbox{digit} & 10 \
frac{ 10 }{ 1 } & 10^2 - 106 cdot 1^2 = -6 & mbox{digit} & 3 \
frac{ 31 }{ 3 } & 31^2 - 106 cdot 3^2 = 7 & mbox{digit} & 2 \
frac{ 72 }{ 7 } & 72^2 - 106 cdot 7^2 = -10 & mbox{digit} & 1 \
frac{ 103 }{ 10 } & 103^2 - 106 cdot 10^2 = 9 & mbox{digit} & 1 \
frac{ 175 }{ 17 } & 175^2 - 106 cdot 17^2 = -9 & mbox{digit} & 1 \
frac{ 278 }{ 27 } & 278^2 - 106 cdot 27^2 = 10 & mbox{digit} & 1 \
frac{ 453 }{ 44 } & 453^2 - 106 cdot 44^2 = -7 & mbox{digit} & 2 \
frac{ 1184 }{ 115 } & 1184^2 - 106 cdot 115^2 = 6 & mbox{digit} & 3 \
frac{ 4005 }{ 389 } & 4005^2 - 106 cdot 389^2 = -1 & mbox{digit} & 20 \
frac{ 81284 }{ 7895 } & 81284^2 - 106 cdot 7895^2 = 6 & mbox{digit} & 3 \
frac{ 247857 }{ 24074 } & 247857^2 - 106 cdot 24074^2 = -7 & mbox{digit} & 2 \
frac{ 576998 }{ 56043 } & 576998^2 - 106 cdot 56043^2 = 10 & mbox{digit} & 1 \
frac{ 824855 }{ 80117 } & 824855^2 - 106 cdot 80117^2 = -9 & mbox{digit} & 1 \
frac{ 1401853 }{ 136160 } & 1401853^2 - 106 cdot 136160^2 = 9 & mbox{digit} & 1 \
frac{ 2226708 }{ 216277 } & 2226708^2 - 106 cdot 216277^2 = -10 & mbox{digit} & 1 \
frac{ 3628561 }{ 352437 } & 3628561^2 - 106 cdot 352437^2 = 7 & mbox{digit} & 2 \
frac{ 9483830 }{ 921151 } & 9483830^2 - 106 cdot 921151^2 = -6 & mbox{digit} & 3 \
frac{ 32080051 }{ 3115890 } & 32080051^2 - 106 cdot 3115890^2 = 1 & mbox{digit} & 20 \
end{array}
$$
answered Jan 31 at 23:17
Will JagyWill Jagy
104k5102201
edited Feb 1 at 19:09
answered Jan 31 at 23:17
Will JagyWill Jagy
104k5102201
answered Jan 31 at 23:17
Will JagyWill Jagy
104k5102201
answered Jan 31 at 23:17
Will JagyWill Jagy
104k5102201
104k5102201
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