# On the Diophantine equations f (x) = g(y)

## TL;DR

This paper investigates conditions under which certain polynomial Diophantine equations have finitely many integer solutions, analyzing specific cases and assuming the ABC Conjecture to establish finiteness criteria.

## Contribution

It provides new criteria for finiteness of solutions to equations of the form f(x) = g(y), including cases with monic polynomials and under the ABC Conjecture.

## Key findings

- Finiteness conditions for (y+q1)...(y+qm)=f(x)
- Finiteness criteria for f(x)=g(y) under ABC Conjecture
- Analysis of polynomial equations with integer solutions

## Abstract

The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for which the Diophantine equatio (y + q1 )(y + q2 ) .... (y + qm ) = f(x) has finitely many solutions in integers. Also assuming ABC Conjecture, we study the conditions for finiteness of integer solutions of the Diophantine equation f(x) = g(y).