Ashleigh Ratcliffe

collaborators

3 papers

math.GM2026

On the shortest open cubic equations

Bogdan Grechuk, Ashleigh Ratcliffe

We use cubic reciprocity to prove that the equation $7 x^{3} + 2 y^{3} = 3 z^{2} + 1$ has no integer solutions. Prior to this work, it was the shortest cubic equation for which the existence of i…

math.GM2026

Integers representable as a difference of two rational fourth powers

Ashleigh Ratcliffe, Tho Nguyen Xuan

In Section 6.6 of the book {\it Number Theory, Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics, Volume 239, Springer (2007)}, Cohen investigated the solubi…

math.NT2025

Generalised Fermat equation: a survey of solved cases

Ashleigh Ratcliffe, Bogdan Grechuk

Generalised Fermat equation (GFE) is the equation of the form $a x^{p} + b y^{q} = c z^{r}$ , where $a, b, c, p, q, r$ are positive integers. If $1/ p + 1/ q + 1/ r < 1$ , GFE is known to have at most finitely…