Kannan Soundararajan: Covering integers using quadratic forms (NTWS 166)
Channel: Number Theory Web Seminar
Date: 16/04/2023

Abstract: How large must $\Delta$ be so that we can cover a substantial proportion of the integers below $X$ using the binary quadratic forms $x^2 +dy^2$ with $d$ below $\Delta$?  Problems involving representations by binary quadratic forms have a long history, going back to Fermat.  The particular problem mentioned here was recently considered by Hanson and Vaughan, and Y. Diao.  In ongoing work with Ben Green, we resolve this problem, and identify a sharp phase transition:  If $\Delta$ is below $(\log X)^{\log 2-\epsilon}$ then zero percent of the integers below $X$ are represented, whereas if $\Delta$ is above $(\log X)^{\log 2 +\epsilon}$ then 100 percent of the integers below $X$ are represented.  

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date:
Thursday, April 6, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 7, 2023 (1am AEST, 3am NZST)