 In addition to the expansions discussed in this article, Ramanujan's sums are used in the proof of Vinogradov's theorem that every sufficiently-large odd number is the sum of three primes.For integers "a" and "b", formula_2 is read ""a" divides "b"" and means that there is an integer "c" such that "b" = "ac".Similarly, formula_3 is read ""a" does not divide "b"".