Three consecutive almost squares

2015/02/28 by Jeremy Rouse, Yilin Yang
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #Elliptic curve #Integer (computer science) #Multiple #Number theory #Point (geometry) #Square-free integer #math.NT #msc:11G05 #msc:11Y50

paper · pdf · doi:10.1142/s1793042116500603

published as Int. J. Number Theory 12 (2016), no. 4, 969-978 · Most difficult case now done using the Heegner point method

arxiv created 2015/07/04 · openalex publication_date 2015/08/24 · openalex created_date 2016/06/24 · arxiv updated 2018/01/22 · openalex updated_date 2026/08/05

Abstract

Given a positive integer [Formula: see text], we let [Formula: see text] denote the squarefree part of [Formula: see text]. We determine all positive integers [Formula: see text] for which [Formula: see text] by relating the problem to finding integral points on elliptic curves. We also prove that there are infinitely many [Formula: see text] for which [Formula: see text].

Citations

• Nonvanishing theorems for L-functions of modular forms and their derivatives
• A classical Diophantine problem and modular forms of weight 3/2
• Heegner points and derivatives ofL-series
• The Arithmetic of Elliptic Curves