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### Solve the Diophantine equation $3x^2 - 2y^2 =1$

Solve $$3x^2 - 2y^2 =1$$ in $\mathbb{Z}$. How can we do it? ( All of answers gave me a great help. Thanks a lot kind stackexchangers.)
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Consider the identity $(b-a)(b+a) = ab - 1$, where $a, b$ are nonnegative integers. We can also express this identity as $a^2 + ab - b^2 = 1$. This identity is clearly true when $a = F_{2i-1}$ and $... 2answers 606 views ### Solving the equation$ x^2-7y^2=-3 $over integers I'd like to solve the following Pell equation: $$x^2-7y^2=-3$$ Where$x$and$y$are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ... 3answers 151 views ### What are some books that are in the spirit of David A. Cox' “Primes of the Form$x^2+ny^2$” David A. Cox "Primes of the Form$x^2+ny^2$: Fermat, Class Field Theory, and Complex Multiplication." has a very good (at least to me, and many) methodology. He starts from page 1 asking a simple ... 3answers 368 views ### Algebraic proof of non-trivial solution to the Pell's equation Let$d$be a square-free positive integer, and consider the pairs$(x, y) \in \mathbb{Z}^2$that satisfy: $$x^2 - dy^2 = 1$$ The existence of a non-trivial solution to this equation (i.e. distinct ... 1answer 722 views ### how to solve binary form$ax^2+bxy+cy^2=m$, for integer and rational$ (x,y)$solve$ 3x^2+3xy-5y^2=55$using number theory tools ,i have found the following$\Delta=3^2+4(5)(3)=9+60=69d=69,u=1w_{69}=\frac{1+\sqrt{69}}{2}O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}...
It is well known that every positive integer is the sum of at most four perfect squares (including $1$). But which positive integers are not the sum of four non-zero perfect squares ($1$ is still ...