MATH Seminar

Title: Equal sums of two cubes of quadratic forms: an apology
Seminar: Algebra
Speaker: Bruce Reznick of University of Illinois at Urbana-Champaign
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-12-04 at 4:00PM
Venue: MSC W301
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Abstract: The topic of equal sums of two cubes has occupied number theorists and algebraists for a long time. In this talk, I will describe a one-parameter family of six binary quadratic forms $f_i$ so that $f_1^3 + f_2^3 = f_3^3 + f_4^3 = f_5^3 + f_6^3$ and so that every pair of equal sums of two cubes arises as one of the equalities here, perhaps with terms flipped. I will name-check Euler, Sylvester and Ramanujan. My favorite single example is \[ (x^2 + x y - y^2)^3 + (x^2 - x y - y^2)^3 = 2x^6 - 2y^6 \] The famous Euler-Binet parameterization of solutions over $\mathbb Q$ will be combined with point-addition of elliptic curve theory in what appears to be a novel way.