10 problems

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

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

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…

A polynomial solution $(\mathcal{A},\mathcal{B},\mathcal{C})$ is constructed for each standard Pythagorean triple, where $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are polynom…

Let $a,b,c$ be relatively prime positive integers satisfying … A solution $(x,y,z)$ in positive integers is nontrivial if it is not $(2,2,2)$. Jesmanowicz's conjecture. The equatio…

Let $b,c,r,t$ be fixed positive integers satisfying … Let $x,y,z\in\mathbb{N}$ satisfy $x^2+b^y=c^z$. Terai's prescribed-exponent conjecture. The only solution is … The conjecture…

Corrected Cao–Dong conjecture. The equation

Let $(a,b,c)$ be a primitive Pythagorean triple with … where $a,b,c\in\mathbb{N}$, $\gcd(a,b)=1$, and $a\equiv0\pmod{2}$. Terai's conjecture. The equation … has only the solution…

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 $(U,V)$…