The Theory of Transfinite Diophantine Equations The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for equations of the form $a_1 x^{n_{1}}_1+\cdots+a_k x^{n_{k}}_k=b y^{m}$ which all of the variables are positive integers.
In this direction if we let variables to vary on transfinite ordinals as generalization of natural numbers, we will enter the realm of transfinite Diophantine equations like $\alpha_1 x^{\beta_{1}}_1+\cdots+\alpha_k x^{\beta_{k}}_k=\gamma y^{\delta}$ which $k$ is a positive integer and all other variables vary on non-zero ordinals. 
It seems there are few known results in the literature about the theory of transfinite Diophantine equations. A possible reason could be the impossibility of calculation on transfinite numbers using super computers which gives an intuition about existence of a solution on integers. Also arithmetic operators lose some of their good properties on transfinite numbers. My questions are about existence of non-trivial solutions for such equations:
Question 1: For which non-trivial transfinite Diophantine equations like $\alpha_1 x^{\beta_{1}}_1+\cdots+\alpha_k x^{\beta_{k}}_k=\gamma y^{\delta}$ do we know that there is no non-trivial solution on both finite and infinite ordinals when $k$ is a positive integer and $\alpha_1,\cdots, \alpha_k,\beta_{1},\cdots, \beta_{k}, \gamma, \delta$ are positive (finite or transfinite) ordinals? 
Question 2: Is there any non-trivial Diophantine equation like $\alpha_1 x^{\beta_{1}}_1+\cdots+\alpha_k x^{\beta_{k}}_k=\gamma y^{\delta}$ with positive integers $k, \alpha_1,\cdots, \alpha_k,\beta_{1},\cdots, \beta_{k}, \gamma, \delta$ which has no non-trivial solution on integers but has a solution on transfinite ordinal numbers?
Question 3: What is known about non-trivial transfinite numbers which can be solutions of the equations like $\alpha_1 x^{\beta_{1}}_1+\cdots+\alpha_k x^{\beta_{k}}_k=\gamma y^{\delta}$ when $k$ is a positive integer and $\alpha_1,\cdots, \alpha_k,\beta_{1},\cdots, \beta_{k}, \gamma, \delta$ are positive (finite or transfinite) ordinals? 
Remark: The above question is similar to searching for Pythagorian triples in the finite case.
Question 4: Is there any geometric intuition about the theory of transfinite Diophantine equations as same as what developed for proving Fermat's conjecture? 
 A: In the abstract of his paper "Concerning a Class of Diophantine Equations with Ordinal Variables", J. L. Hickman says:

It has been shown by Sierpiński that the Diophantine equation $x^3 = y^3 + 1$ has no solution in transfinite ordinals. A short time later, Świerczkowski generalized this result by demonstrating the non-existance of transfinite solutions of the equation $x^α = y^{rβ} + k$, where $k$ is a positive integer and $α,rβ$ are successor ordinals $> 1$. We commence this note with a simple demonstration that the restriction of $α,rβ$ to successor ordinals was unnecessary, and then go on to consider the existence or non-existence of transfinite solution of a related class of Diophantine equations. 

A: I note that the equations arising in Fermat's last theorem $$x^n+y^n=z^n$$ all have nonzero solutions in the ordinals, since for any positive natural number $n$ we have $$17^n+\omega^n=\omega^n, $$ where $17$ is any finite natural number. This solution uses the usual ordinal arithmetic, but it may be interesting to consider the question with the alternative natural ordinal arithmetic, which is commutative, and these solutions no longer work. Is FLT provable in the ordinals for the commutative ordinal addition? 
