Intereting Posts

Proving that the dual of the $\mathcal{l}_p$ norm is the $\mathcal{l}_q$ norm.
Sum of odd Fibonacci Numbers
Reference request: is mathematics discovered or created?
Intersection Number of $B = Y^2 – X^3 + X$ and $F = (X^2 + Y^2)^3 – 4X^2Y^2$ using the fact $I(P,F \cap B) = ord_P^B(F) $.
$2+2 = 5$? error in proof
Do odd imaginary numbers exist?
Using Vieta's theorem for cubic equations to derive the cubic discriminant
Sequence of measurable functions
How to construct a line with only a short ruler
101 positive integers placed on a circle
Difference between “$G$ acts on $A$” and “G is a permutation group on $A$ (i.e. $G\leq S_A$)”
Properties and notation of third-order (and higher) partial-derivatives
Homology groups of a tetrahedron
Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
Is $\left(\sum\limits_{k=1}^\infty \frac{x_k}{j+k}\right)_{j\geq 1}\in\ell_2$ true if $(x_k)_{k\geq 1}\in\ell_2$

Just trying to figure out a way to generate triples for $a^2+b^2=5c^2$. The wiki article shows how it is done for $a^2+b^2=c^2$ but I am not sure how to extrapolate.

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- Geometric interpretation and computation of the Normal bundle
- Does the Pell-like equation $X^2-dY^2=k$ have a simple recursion like $X^2-dY^2=1$?
- On the definition of the structure sheaf attached to $Spec A$
- Vakil 14.2.E: $L\approx O_X(div(s))$ for s a rational section.
- 4-ellipse with distance R from four foci
- The twisted cubic is an affine variety.
- Complex analysis book for Algebraic Geometers
- $x^5 + y^2 = z^3$

This is one of those CW answers. Country and Western.

I did Gerry’s recipe and I quite like how it works. Educational, you might say.

I took the slope $t = \frac{q}{r}$ and starting rational point $(2,1).$ The other point works out to be

$$ x = \frac{2 t^2 – 2 t – 2}{t^2 + 1}, \; \; \; y = \frac{- t^2 – 4 t + 1}{t^2 + 1}, $$ so multiply everything by $r^2$ to arrive at

$$ x = \frac{2 q^2 – 2qr – 2r^2}{q^2 + r^2}, \; \; \; y = \frac{- q^2 – 4 qr + r^2}{q^2 + r^2}. $$

So far $x^2 + y^2 = 5.$ Multiply through by $q^2 + r^2$ to get

$$ a = 2 q^2 – 2 q r – 2 r^2 $$

$$ b = -q^2 – 4 q r + r^2 $$

$$ c = q^2 + r^2 $$

$$ a^2 + b^2 = 5 q^4 + 10 r^2 q^2 + 5 r^4 $$

and

$$ c^2 = q^4 + 2 r^2 q^2 + r^4 $$

and

$$ a^2 + b^2 = 5 c^2 $$

Consider the circle $$x^2+y^2=5$$ Find a rational point on it (that shouldn’t be too hard). Then imagine a line with slope $t$ through that point. It hits the circle at another rational point. So you get a family of rational points, parametrized by $t$. Rational points on the circle are integer points on $a^2+b^2=5c^2$.

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- Prime Partition
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