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Almost Pell type equation

If you multiply this by $2$ you get $(2x)^2 - 2N y^2 = -2$, so you can solve $x^2 - 2N y^2 = -2$ and then check when $x$ is even. $x^2 - 2Ny^2 = -2$ is a generalized Pell's equation, so you can find a ...

Another generalisation of euclidean division on integers

Even if we have only $f(m+k)=g(f(m),f(k))$, these functions may be classified as follows. It is more convenient to consider not the function $f$, but the corresponding equivalence relation: $a\sim b$ ...
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Picard group of quasi-projective varieties

The paper R. Guralnick, D. B. Jaffe, W. Raskind, and R. Wiegand, On the Picard group: Torsion and the kernel induced by a faithfully flat map, J. Algebra, vol. 183, no. 2, pp. 420–455, 1996. DOI: 10....
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When $E_D:y^2=x^3+17D^2x$ has even rank?
You can compute the root number over $\mathbb{Q}$ of any elliptic curve of the form $y^2=x^3+\alpha x$ for $\alpha\in \mathbb{Q}^*$ using the formulae in section 4B in Density of rational points on ...