Due to the efforts of Fermat, Euler, Legendre and Gauss, it is known what
natural numbers can be written as the sum of two squares or three squares.
Lagrange's four-square theorem proved in 1770 states that each natural
number can be expressed as the sum of four squares.
In the talk we will first review classical results on sums of two or three
or four squares. Then we turn to the speaker's recent discoveries which
refine Lagrange's four-square theorem or the Gauss-Legendre theorem on
sums of three squares.
In particular we will introduce our results refining Lagrange's
four-square theorem as well as some recent conjectures of the speaker one
of which states that any integer greater than one can be written as the
sum of two squares, a power of 3 and a power of 5.