(Moved from previous post since the answers were not elliptic curves.) I. Question We seek to find infinitely many primitive solutions to, $$a^4+b^4+c^4 = d^4+e^4$$ where $d \color{red}{\ne} e$ using polynomial solutions or elliptic curves. The most well-known case when $d = e$ is,...

(Updated with a computer search.) I. Question We seek to find infinitely many primitive solutions to, $$a^4+b^4+c^4 = d^4+e^4$$ where $d \color{red}{\ne} e$. The most well-known case when $d = e$ is, $$a^4+b^4+(a+b)^4 = 2(a^2+ab+b^2)^2$$ where one then solves $a^2+ab+b^2 = z^k$ for $k=2$. (In...

Let $S\subset \mathbb R^3$ be a fixed sphere and let $E\subset \mathbb R^3$ be a fixed ellipsoid containing $S$. Consider tetrahedra $$A_1A_2A_3A_4$$ such that $$A_i\in E$$ and each face is tangent to $S$. Let $D_i\in S$ be the tangency point of the face opposite $A_i$. Poncelet’s closure...

The curve $y^2=P(x)$ over the field of complex numbers, where $P(x)$ is a polynomial of degree $4$ without repeating roots, can be transformed with a birational transformation into $Y^2=Q(X)$ with $Q(x)$ of degree $3$ without repeating roots. That is, an elliptic curve. However, if $P(x)$ does...

$$3(a^3+b^3+c^3+a+b+c)+5(a^2+b^2+c^2)+2(a^2b+b^2c+ac^2)+4(a^2c+bc^2+ab^2+ab+bc+ac)=0$$ A table of primitives known to me $(a, b, c)$, $a \in{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}^+$ $$ \boxed{\begin{array} {|r|r|r|r|}\hline №(a,b,c)& a_n & b_n & c_n \\ \hline S_1 & 0 & 0 & 0 \\ \hline S_2 & 0...

Let $$ \mathcal S=(\mathcal I,\Gamma,\Pi) $$ be a seam marked seed built from four compact oriented $2$-dimensional complex orbifold sheets. Each sheet is assumed to be a football type orbifold: its coarse underlying Riemann surface is $$ |\mathcal O_i|\cong \mathbb P^1 $$ and it has two...

In a prior post, the equation, $$x_1^5+2x_2^5 = y_1^5+2y_2^5$$ was considered. It has only one known primitive solution. This present post considers the similar, $$x_1^k+x_2^k+x_3^k = y_1^k+2y_2^k$$ valid for both $k = (1,5)$. Duncan Moore found only one primitive solution, namely, $$85333^k +...

I have a question about a certain part of the proof of Theorem 4.2 from Rational Points on Elliptic Curves. Let $p$ be a prime so that $p=1\;\mathrm{mod}\;3$. Let $R=\{x^{3}|x\in\mathbb{F}_{p},x\neq 0\}$. Notation: $[XYZ]$ is the number of triples $(x,y,z)$ so that $x+y+z=0,x\in X,y\in Y,z\in...

I’m working on a paper in which I end up considering the biquadratic rational curve $$u^2v^2 - u^2 - v^2 - 6uv + 8 = 0. \tag{$1$}$$ To complete the remainder of my proof/method, I need to prove that it has rank 1. I believe it can be transformed to the Weierstrass form $$Y^2=X^3-32X,$$ and then...

The above given equation solution is very easy just make it to an elliptic curve. For $$ W^{4}+X^{4}=Y^{2}+Z^{4} $$ Divide both sides $Z^{4}$, we wil get $$ \left(\frac {W}{Z}\right)^4 + \left(\frac{X}{Z}\right)^4 = \left(\frac{Y}{Z^2}\right)^2 +1 $$ If we substitute $\frac{W}{Z} = (u+v)$,...

Problem:I am looking for help with the following Diophantine equation: $$y^2 = x^3 - x^2 + 16$$ By working through the equation, I have successfully found 8 distinct non- negative integer solutions. The largest value of $x$ among all the solutions I found is $x = 112$ (which gives $y = 1180$)....