An interesting MSE post was recently made by Koushik Pramanik. To give some background, there seems to be only one known polynomial solution to the equation in the first part of the title, namely, $$(17 p^2 - 12 p q - 13 q^2)^4 + (17 p^2 + 12 p q - 13 q^2)^4 = (17 p^2 - q^2)^4 + (289 p^4 + 14...

In 2008, Jacobi-Madden found (essentially by data-mining the 25 smallest solutions) that $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 =e^4$$ was solvable and in fact a member of an infinite family. In early August 2026, Matej Veselovac and I were data-mining the first 45,000 smallest solutions of Eugene Go's...

A while ago, @Aleksandr posed this question, about finding new solutions to $$a^8+b^8+c^8+d^8=e^8+f^8+g^8+h^8$$ (where the solutions should be non-trivial and primitive) and he stated the known result that in 2006, Nuutti Kuosa discovered...

It is well known that a parameterization for the equation $$W^2+X^2=Y^2+Z^2$$ is given by: $$(W,X,Y,Z)=(a+b,ab-1,ab+1,a-b).$$ I have found a similar type of parameterization for the fourth-power equation: $$W^4+X^4=Y^4+Z^4$$ where $W = d+e$, $X = cde-1$, $Y = d-e$, and $Z = cde+1$, provided that...

(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 $y = 1180$)....