Some Ideas for Final Projects

1.8. Investigate algorithms for expressing an integer as a sum of squares. We have determined exactly which integers are sums of two squares. Later on in the course we will prove which integers are sums of four squares (Lagrange’s theorem) and state without proof which integers are sums of three squares (Legendre-Gauss theorem). But these methods do not give efficient algorithms for actually finding any of these representations. E.g, if p ≡ 1 (mod 4) is a large prime number, we know that there exist x, y ∈ Z such that p = x + y? There is an obvious trial and error approach to finding x and y: compute p− 1, p− 2, . . . until we get a perfect