Question

Number theory: Part C and Part D please!

QUADRA range's Four-Square Theorem) If n is a natural be expressed as the sum of four squares. insmber, then n cam be expressed tice Λ in 4-space is a set of the form t(x,y, z, w). M:x,y,z, w Z) matrix of nonzero determinant. The covolume re M is a 4-by-4 no is defined to be the absolute value of Det M such a lattice, of covolume V, and let S be the 4-sphere et Ab 4 24 s of radius r.3s Minkowski's theorem in 4 x2 + y2 + z2 + w2-p2 dimensions states that if Λ be s The volu 224. V, 2 then the sphere contains a non-origin point of the lattice. Let p be an odd prime. Prove that there exist integers r,s such that? + s2 =-1 mod p. Hint: consider the equivalent congru- ence r2 +1 -s2 mod p. As r and s vary, why must the value on the left be congruent to the value on the right, at least once? +1 mod p. Consider the lattice 0o) Suppose that "Si m (px + rz + sto, py + sz-rw, z, w) : x, y, z, w E Z). Л exist Prove that if (a,b,c,d) EA,then a2 + b2 + c2 + d2-0mod p (e) Use Minkowski's theorem to prove that there exist a, b,c, d such that a+b2+c+dp. (d) In a 1748 letter to Goldbach, Euler wrote the text in the margin. Now prove Lagrange's Four-Square Theorem. nd a is an integer,

Answer 1

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