Sum of two squares theorem

The prime decomposition of the number 2450 is given by 2450 = 257. Of the primes occurring in this decomposition, 2, 5, and 7, only 7 is congruent to 3 modulo 4. Its exponent in the decomposition, 2, is even. Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, . The prime decomposition of the number 3430 is 257. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum of two squares. ==Representable numbers==

The numbers that can be represented as the sums of two squares form the integer sequence :0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, ... They form the set of all norms of Gaussian integers; The product of any two representable numbers is another representable number. Its representation can be derived from representations of its two factors, using the Brahmagupta–Fibonacci identity. == Jacobi's two-square theorem ==

{{Math theorem Denote the number of divisors of n as d(n), and write d_a(n) for the number of those divisors with d \equiv a \pmod 4. Let n = 2^f p_1 ^{r_1} p_2 ^ {r_2} \cdots q_1^{s_1} q_2^{s_2} \cdots where p_i \equiv 1 \pmod 4, \ q_i \equiv 3 \pmod 4. Let r_2(n) be the number of ways n can be represented as the sum of two squares. Then, r_2(n) = 0 if any of the exponents s_j are odd. If all s_j are even, then r_2(n) = 4 d (p_1^{r_1} p_2^{r_2} \cdots ) = 4(d_1(n) - d_3(n)) }} Proved by Gauss using quadratic forms and Jacobi using elliptic functions. An elementary proof is based on the unique factorization of the Gaussian integers. == See also ==