An effective estimate for the sum of two cubes problem

2024/03/11 by Saunak Bhattacharjee, Bhattacharjee, Saunak
Engineering · #11G50 #FOS: Mathematics #Number Theory (math.NT) #Optimization and Packing Problems #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2403.17955

openalex publication_date 2024/03/11 · openalex created_date 2024/03/29 · openalex updated_date 2026/07/28

Abstract

Let f(x, y) ∈ ℤ[x, y] be a cubic form with non-zero discriminant, and for each integer m ∈ ℤ, let, N_f(m)=#\(x, y) ∈ ℤ²: f(x, y)=m\ . In 1983, Silverman proved that N_f(m)>Ω((log |m|)^{3 / 5}) when f(x, y)=x³+y³. In this paper, we obtain an explicit bound for N_f(m), namely, showing that N_f(m)>4.2× 10^-6(log |m|)^11/13 (holds for infinitely many integers m), when f(x, y)=x³+y³.

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