Decomposition of a Number as a Sum of Two or Four Squares

The nature of a prime capable of expression as a sum of the form x ² + y ² or x ² + 2y ² or x ² + 3y ² or x ² + 7y ² is examined. Fermat’s two-squares theorem about primes of the form 4k + 1 is proved using the norm of an element in ℤ[i], the integral domain of Gaussian integers. A theorem of Edmund Landau (1877–1938) on the expressibility of a number r (> 1) as a sum of three squares says: If r is a sum of three squares, then, it is not of the form 4 ^a (8b + 7). a ≥ 0, b ≥ 0.