Tohoku Mathematical Journal
2026
June
SECOND SERIES VOL. 78, NO. 2

Tohoku Math. J.
78 (2026), 233-241

Title INTEGERS THAT ARE SUMS OF TWO CUBES IN THE CYCLOTOMIC $\mathbb{Z}_p$-EXTENSION

Author Anwesh Ray

(Received May 24, 2024, revised September 27, 2024)
Abstract. Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation $n=x^3+y^3$, where $x$ and $y$ belong to the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, consider the case when $n$ is not a sum of rational cubes. Then, we prove that $n$ cannot be a sum of two cubes in certain large families of prime cyclic extensions of $\mathbb{Q}$.

Mathematics Subject Classification. Primary 11R23, Secondary 11G05, 11D25.
Key words and phrases. Ranks of elliptic curves, diophantine applications of Iwasawa theory, representing a number as a sum of two cubes.

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