Integers representable as a difference of two rational fourth powers

Ashleigh Ratcliffe; Tho Nguyen Xuan

Integers representable as a difference of two rational fourth powers

General Mathematics 2026-04-28 v2

Authors: Ashleigh Ratcliffe , Tho Nguyen Xuan

Abstract

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 solubility of the equation $n=x^4+y^4$ in the rational numbers $x,y$ for all positive integers $n \leq 10000$ . Motivated by this, we investigate the equation $n=x^4-y^4$ and obtain the complete list of positive integers $n\leq 10000$ that can be represented in this form for some nonzero rational numbers $x$ and $y$ .

Keywords

number theory perfect numbers

Cite

@article{arxiv.2604.15832,
  title  = {Integers representable as a difference of two rational fourth powers},
  author = {Ashleigh Ratcliffe and Tho Nguyen Xuan},
  journal= {arXiv preprint arXiv:2604.15832},
  year   = {2026}
}

Integers representable as a difference of two rational fourth powers

Abstract

Keywords

Cite

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