---
title: "Integers expressible as the sum of two rational cubes"
authors:
  - "Levent Alpoge"
  - "Manjul Bhargava"
  - "Ari Shnidman"
year: 2022
venue: "arXiv preprint"
arxiv: "2210.10730"
doi: null
source: "https://arxiv.org/abs/2210.10730"
topics:
  - "arithmetic-geometry"
  - "number-theory"
  - "elliptic-curves"
seed_rank: 1201
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 10
lineage: arithmetic-geometry
cites:
  - title: "Quadrics in arithmetic statistics"
    url: "https://arxiv.org/abs/2110.03947"
    year: 2021
    arxiv: "2110.03947"
    doi: null
see:
  []
---

# Integers expressible as the sum of two rational cubes

## One-sentence takeaway

A positive proportion of integers are sums of two rational cubes, and a positive proportion are not — Davenport's conjecture — by proving that average 2-Selmer in any cubic-twist family, at fixed root number, is 3.

## Why it's lovely

x^3 + y^3 = n over Q, the Mordell curve cubic-twist family. Geometry-of-numbers plus circle method on pairs of binary cubics (Bhargava-Ho), with the singular integral/series reread as real and p-adic Haar measure. Quanta covered it.

## Problem

Davenport: which integers are sums of two rational cubes? Expected density positive and complementary. Same for squares.

## Design

Parametrize 2-Selmer by pairs of binary cubics; count on the quadric via circle method plus uniformity; sieve to fixed root number; apply p-parity (Dokchitser-Dokchitser) and a p-converse (Burungale-Skinner). Rank 0 at least 1/6 of any cubic-twist family; rank 1 at least 1/6 of those with good reduction at 2.

## Evidence

arXiv:2210.10730, v3 19 Oct 2024. No journal id found at ingest — do not invent.

## Limitations

Still a preprint. Average Selmer, not a rank distribution. Relies on the p-converse at 2.

## Implications for Broadside

The circle-method-in-arithmetic-statistics paper the public knows. Method paper is 1204.

## Bottom line

Mint 2210.10730.

## Links

- arXiv: [2210.10730](https://arxiv.org/abs/2210.10730)