Migrating to this repo because this repository is very crowded and messy with many files and folders. This repo aims to store/contain attempts (solutions at best) to this catalogue of open polynomial diophantine problems from Version 7 of this paper. Every folder contains a (potential) solution to an open problem from the aforementioned paper. Efforts to verify the correctness of this paper include Lean formalisations with no sorry's (with Aristotle) found in every folder and I will soon include in each folder links leading to three different windows of ChatGPT 5.5 Pro where I stress-test the correctness of these results as an extra layer of verification.

Folder	Goal	Progress
[1]	Prove the infinitude of integer solutions for the equation: $y^2 + x^3 \cdot y + z^2 + 1 = 0$ (Table 13, Problem 4).	✅ Success!
[2]	Prove the infinitude of integer solutions for the equation: $y^2 + x^3 \cdot y + z^2 - 2 = 0$ (Table 13, Problem 4).	(Already solved)
[3]	Prove the infinitude of integer solutions for the equation: $z^2 + y^2 \cdot z+2x^3 + 1=0$ (Table 13, Problem 4).	✅ Success!
[4]	Prove that the equation: $y(x^3-z^2)=x$ has no positive integer solutions (Problem 7).	🔴 Still needs working on
[5]	Prove that the equation: $z^2 + y^2 \cdot z + x^3 - 2 = 0$ has infinitely many integer solutions (Table 13, Problem 4).	🔴 Still needs working on
[6]	Prove that the equation: $y^2 + x^2 \cdot y + z^2 \cdot x + 1 = 0$ has infinitely many integer solutions (Table 13, Problem 4).	(Already solved)
[7]	Prove that the equation: $y(x³ − z²) = z + 1$ has infinitely many integer solutions (Table 13, Problem 4).	(Already solved)
[8]	Prove that the equation: $y(x³ − z²) = x - 1$ has infinitely many integer solutions (Table 13, Problem 4).	(Already solved)
[9]	Describe all integer solutions of the equation: $y^2+x^2y+xz^2-2=0$ (Table 5, Problem 2).	(Already solved)
[10]	Describe all integer solutions of the equation: $z^2-xy^2-x^3-2=0$ (Table 5, Problem 2).	(Already solved)
[11]	Describe all integer solutions of the equation: $z^2-xy^2-x^3+2=0$ (Table 5, Problem 2).	(Already solved)
[12]	Describe all integer solutions of the equation: $y^2+x^2y+xz^2+2=0$ (Table 5, Problem 2).	(Already solved)
[13]	Describe all integer solutions of the equation: $y^2 + x^2 \cdot y + z^2 \cdot x + 1 = 0$ (Table 5, Problem 2).	(Already solved)
[14]	Find all primitive integer solutions to the equation: $3Y^3+Z^3=X^4$ (MathOverflow).	🟡 Pending feedback

Since Version 8 (the latest) of this paper, these problem(s) have been solved:

Folder	Goal	Progress
[15]	Prove that the equation: $x^3 + x^2 \cdot y^2+z^2 + 1=0$ has infinitely many integer solutions (Table 12).	✅ Success!
[16]	Describe all rational/integer solutions to the equation: $y^2 - x^2 \cdot y + z^2 + 1 = 0$.	🔴 Still needs working on
[17]	Provide a polynomial parametrisation to the equation: $xyz + ts + 1 = 0$.	🔴 Still needs working on
[18]	Provide a polynomial parametrisation to the equation: $xyz + t^2 + 1 = 0$.	🔴 Still needs working on
[19]	Provide a polynomial parametrisation to the equation: $xyz + t^2 - 1 = 0$.	🔴 Still needs working on
[20]	Provide a polynomial parametrisation to the equation: $x^3 + yz + 1 = 0$.	🔴 Still needs working on

Since Version 9 (the latest) of this paper, these problem(s) have been solved:

Folder	Goal	Progress
[21]	Prove that the equation: $y^3+y^2+xy=x^4+1$ has no integer solutions (Table 9).	(Already solved)
[22]	Prove that the equation: $y^3+xy=x^4+x+3$ has no integer solutions (Table 9).	(Already solved)

For the problems marked (Already solved), see the latest version of Dr. Grechuk's (aforementioned) paper. I used GPT-Codex to build a simple website to track the progress of solved equations in the aforementioned paper here. This is its GitHub repo.

The table below document efforts towards (partially or unconditionally) solving problems from Table 1 of the joint paper between Dr. Grechuk and myself.

Folder	Goal	Status
[A]	Prove the finiteness of integer solutions for the equation: $y (x^3 - y^2 ) = 2x - 1$.	🟡 Pending feedback (Will be revisited)
[B]	Prove the finiteness of integer solutions for the equation: $x^4+xy+y^3-1=0$.	🟡 Pending feedback
[C]	Prove the finiteness of integer solutions for the equation: $x^4+xy+y^3+1=0$.	🟡 Pending feedback
[D]	Prove that the equation: $y^3 - y = x^4 - x$ has finitely many integer solutions.	🟡 Pending feedback
[E]	Prove that the equation: $y^3 + y = x^4 + x$ has finitely many integer solutions.	🟡 Pending feedback
[F]	Prove that the equation: $x^3 y^2 = z^4 + 1$ has finitely many integer solutions.	🟡 Pending feedback
[G]	Prove that the equation: $x^4 y^3 = z^2 + 1$ has finitely many integer solutions.	🟡 Pending feedback
[H]	Prove that the equation: $x^3 y^2 = 2z^3 + 1$ has finitely many integer solutions.	🟡 Pending feedback
[I]	Describe all integer solutions of the equation: $y^2+x^2y+xz^2-2=0$ (Table 5, Problem 2).	(Already solved)
[J]	Describe all integer solutions of the equation: $z^2-xy^2-x^3-2=0$ (Table 5, Problem 2).	(Already solved)
[K]	Describe all integer solutions of the equation: $z^2-xy^2-x^3+2=0$ (Table 5, Problem 2).	(Already solved)
[L]	Describe all integer solutions of the equation: $y^2+x^2y+xz^2+2=0$ (Table 5, Problem 2).	(Already solved)
[M]	Describe all integer solutions of the equation: $y^2 + x^2 \cdot y + z^2 \cdot x + 1 = 0$ (Table 5, Problem 2).	(Already solved)
[N]	Find all primitive integer solutions to the equation: $3Y^3+Z^3=X^4$ (MathOverflow).	🟡 Pending feedback