12 problems

Jeśmanowicz's conjecture. The Diophantine equation

Erdős–Graham conjecture. Any finite colouring of $\mathbb{N}$ has a monochromatic Pythagorean triple.

Ergodic Pythagorean recurrence conjecture. There exists a solution $(x,y,z)\in\mathbb{N}^3$ of $x^2+y^2=z^2$ such that

Let $\mathcal{G}$ be the graph of Pythagorean walks, with distinguished vertex $O(0,0)$, and let graph distance be measured from $O(0,0)$. A vertex is a symmetric counterpart of a…

The trivial-triple conjecture. If this system has a solution, then $(x,y,z)$ or $(u,v,w)$ equals $(1,0,1)$. A counterexample would yield a perfect Euler cuboid, with edges and face…

Let $(x,y,z)$ and $(u,v,w)$ be primitive Pythagorean triples. The system … has a solution only if $(x,y,z)$ or $(u,v,w)$ equals $(1,0,1)$. The trivial-triple conjecture. If the sys…

Corrected Cao–Dong conjecture. The equation

Cao–Dong's conjecture. The equation

Infinitude conjecture. There are infinitely many primitive Pythagorean triples $(a,b,c)$ for which $E_{a,b}$ has rank zero.

Miyazaki's shuffle conjecture. If $f=g+1$, the only solution is $(x,y,z)=(1,1,2)$; otherwise, there are no solutions. The source describes this as unsolved, with verification in se…

Consider the trinary trees generated by $(2,1)$ and $(3,1)$. For a relatively prime pair $(m,n)$, let $(u,v)$ be the pair in a Bézout tree corresponding to $(m,n)$, and let $(U,V)$…

All-palindromic triple conjecture. The triple $(3,4,5)$ is the only primitive Pythagorean triple whose three components are numeric palindromes.