# 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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