Bdtyktl

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.

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 $x$.

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 $pm 1.$

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:

$$ sqrt { 106} = 10 + frac{ sqrt {106} - 10 }{ 1 } $$
$$ frac{ 1 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{6 } = 3 + frac{ sqrt {106} - 8 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{7 } = 2 + frac{ sqrt {106} - 6 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{10 } = 1 + frac{ sqrt {106} - 4 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{9 } = 1 + frac{ sqrt {106} - 5 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 5 } = frac{ sqrt {106} + 5 }{9 } = 1 + frac{ sqrt {106} - 4 }{9 } $$
$$ frac{ 9 }{ sqrt {106} - 4 } = frac{ sqrt {106} + 4 }{10 } = 1 + frac{ sqrt {106} - 6 }{10 } $$
$$ frac{ 10 }{ sqrt {106} - 6 } = frac{ sqrt {106} + 6 }{7 } = 2 + frac{ sqrt {106} - 8 }{7 } $$
$$ frac{ 7 }{ sqrt {106} - 8 } = frac{ sqrt {106} + 8 }{6 } = 3 + frac{ sqrt {106} - 10 }{6 } $$
$$ frac{ 6 }{ sqrt {106} - 10 } = frac{ sqrt {106} + 10 }{1 } = 20 + frac{ sqrt {106} - 10 }{1 } $$

Simple continued fraction tableau:
$$
begin{array}{cccccccccccccccccccccccccccccccccccccccc}
& & 10 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & & 3 & & 2 & & 1 & & 1 & & 1 & & 1 & & 2 & & 3 & & 20 & \
\
frac{ 0 }{ 1 } & frac{ 1 }{ 0 } & & frac{ 10 }{ 1 } & & frac{ 31 }{ 3 } & & frac{ 72 }{ 7 } & & frac{ 103 }{ 10 } & & frac{ 175 }{ 17 } & & frac{ 278 }{ 27 } & & frac{ 453 }{ 44 } & & frac{ 1184 }{ 115 } & & frac{ 4005 }{ 389 } & & frac{ 81284 }{ 7895 } & & frac{ 247857 }{ 24074 } & & frac{ 576998 }{ 56043 } & & frac{ 824855 }{ 80117 } & & frac{ 1401853 }{ 136160 } & & frac{ 2226708 }{ 216277 } & & frac{ 3628561 }{ 352437 } & & frac{ 9483830 }{ 921151 } & & frac{ 32080051 }{ 3115890 } \
\
& 1 & & -6 & & 7 & & -10 & & 9 & & -9 & & 10 & & -7 & & 6 & & -1 & & 6 & & -7 & & 10 & & -9 & & 9 & & -10 & & 7 & & -6 & & 1
end{array}
$$

$$
begin{array}{cccc}
frac{ 1 }{ 0 } & 1^2 - 106 cdot 0^2 = 1 & mbox{digit} & 10 \
frac{ 10 }{ 1 } & 10^2 - 106 cdot 1^2 = -6 & mbox{digit} & 3 \
frac{ 31 }{ 3 } & 31^2 - 106 cdot 3^2 = 7 & mbox{digit} & 2 \
frac{ 72 }{ 7 } & 72^2 - 106 cdot 7^2 = -10 & mbox{digit} & 1 \
frac{ 103 }{ 10 } & 103^2 - 106 cdot 10^2 = 9 & mbox{digit} & 1 \
frac{ 175 }{ 17 } & 175^2 - 106 cdot 17^2 = -9 & mbox{digit} & 1 \
frac{ 278 }{ 27 } & 278^2 - 106 cdot 27^2 = 10 & mbox{digit} & 1 \
frac{ 453 }{ 44 } & 453^2 - 106 cdot 44^2 = -7 & mbox{digit} & 2 \
frac{ 1184 }{ 115 } & 1184^2 - 106 cdot 115^2 = 6 & mbox{digit} & 3 \
frac{ 4005 }{ 389 } & 4005^2 - 106 cdot 389^2 = -1 & mbox{digit} & 20 \
frac{ 81284 }{ 7895 } & 81284^2 - 106 cdot 7895^2 = 6 & mbox{digit} & 3 \
frac{ 247857 }{ 24074 } & 247857^2 - 106 cdot 24074^2 = -7 & mbox{digit} & 2 \
frac{ 576998 }{ 56043 } & 576998^2 - 106 cdot 56043^2 = 10 & mbox{digit} & 1 \
frac{ 824855 }{ 80117 } & 824855^2 - 106 cdot 80117^2 = -9 & mbox{digit} & 1 \
frac{ 1401853 }{ 136160 } & 1401853^2 - 106 cdot 136160^2 = 9 & mbox{digit} & 1 \
frac{ 2226708 }{ 216277 } & 2226708^2 - 106 cdot 216277^2 = -10 & mbox{digit} & 1 \
frac{ 3628561 }{ 352437 } & 3628561^2 - 106 cdot 352437^2 = 7 & mbox{digit} & 2 \
frac{ 9483830 }{ 921151 } & 9483830^2 - 106 cdot 921151^2 = -6 & mbox{digit} & 3 \
frac{ 32080051 }{ 3115890 } & 32080051^2 - 106 cdot 3115890^2 = 1 & mbox{digit} & 20 \
end{array}
$$