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Abstract
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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 recent progress on the speaker's 1-3-5 conjecture which states that each natural number can be written as x^2+y^2+z^2+w^2 with x,y,z,w nonnegative integers such that x+3y+5z is a square.
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