By the
sum of two squares theorem, the numbers that can be expressed as a sum of two squares of integers are the ones for which each
prime number congruent to 3 mod 4 appears with an even exponent in their
prime factorization. For instance,
45 = 9 + 36 is a sum of two squares; in its prime factorization, 32 × 5, the prime 3 appears with an even exponent, and the prime 5 is congruent to 1 mod 4, so its exponent can be odd. Landau's theorem states that if N(x) is the number of positive integers less than x that are the sum of two squares, then :\lim_{x\rightarrow\infty}\ \left(\dfrac{N(x)}{\dfrac{x}{\sqrt{\log(x)}}}\right)=b\approx 0.764223653589220662990698731250092328116790541 , where b is the Landau–Ramanujan constant. The Landau-Ramanujan constant can also be written as an infinite product: b = \frac{1}{\sqrt{2}}\prod_{p\equiv 3 \pmod{4}} \left(1 - \frac{1}{p^2}\right)^{-1/2} = \frac{\pi}{4} \prod_{p\equiv 1 \pmod{4}} \left(1 - \frac{1}{p^2}\right)^{1/2}. ==History==