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computational quantity principle – A Pell affection equation retort

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computational quantity principle – A Pell affection equation

If one takes in widespread $(star), ,x^2-dy^2=C$ the place $d$, $C$ in $mathbb{N}$.

Taking $d=w^2p^2+p$ with $earn mathbb{Q}ge 1$ and $pin mathbb{Z}$ which is verified (defined later), for the matrix $$A=commence{pmatrix}2w^2p+1&2w(w^2p^2+p)2w&2w^2p+1cease{pmatrix}$$ if $X_0$ is an answer to $(star)$ then $AX_0$ is one other one.

Now $w$ may breathe taken in a artic route basicaly say $d=a^2b^2+cb$ with $cinmathbb{Z}$, $|c|<|a|$ and $c$ coprime with $b$ and $a$, letting
$w=frac{a}{c}$ and $p=cb$ the matrix $A$ is in $mathbb{Q}$ however can maintain an influence $A^n$ with integer entries. So to say that i did not discover any reference for this thought which is stunning. This is said and identified of passage as a Pell equation when $earn mathbb{N}$.

A query is that if there’s a associated theme dialogue to this method since Pell equations are identified, and as a surmise to provide inescapable household of $A$ with $A^n$ of integer entries. (It seems there are a lot of). Thanks

Edit, i am going to illustrate this in an occasion only for readability:
$$x^2-2021y^2=d^2$$ one resolution is $(d,0)$, i took $2021$ by random as it’s inside what i can purchase, (i do not know if it ought to labor for $2020$) since $2021=frac{45^2}{4^2}4^2-4$. An unostentatious dispute says if the numerator of $w$ right here $45$ is $5 pmod{8}$ Then $A^3in mathbb{M}_2(mathbb{Z})$ so

$$A=commence{pmatrix}-1011.5&45472.522.5&-1011.5cease{pmatrix}$$ and $$A^3=commence{pmatrix}-4.139590049times 10^9&1.8609747948times 10^{11}9.2081880times 10^7&-4.139590049times 10^9cease{pmatrix}.$$

Edit. It appears such $A$ has an all integer energy $A^n$ if and provided that $c$ is an influence of two and primarily $|c|= 1, 2, 4$,

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