# Undergraduate Seminar: "Sums of Two Sixth Powers."

Speaker: Alexis Newton, Emory University

*Abstract: What is the smallest positive integer that is a sum of two rational sixth powers, but not a sum of two integer sixth powers? Consider some positive integer m and look for a solution to a^6+b^6 is equivalent to 0 mod m^6. If (x,y) is a nonzero solution to this equivalence, then (x/m)^6+(y/m)^6 is an integer. Then using a lattice reduction algorithm, we can scale x and y by some c to ensure cy and cx are "small mod 5^6". This gives the possible solution 164,634,913. In this talk, we will discuss the tools used to determine whether 164,634,913 is the smallest positive integer that is a sum of two rational sixth powers, but not a sum of two integer sixth powers. *

*Host: Adeli Hutton*