Littlewood's conjecture is a hypothesis in number theory proposed by mathematician J. E. Littlewood. It suggests that for any two real numbers, the product of their fractional parts will be close to zero infinitely often. More formally, it states that for any two real numbers x and y , the expression \nx\ \ny\ approaches zero as n increases, where \.\ denotes the fractional part. The conjecture is significant in the study of Diophantine approximation and has implications for understanding the distribution of fractional parts of sequences. Despite its simplicity, it remains unproven, making it an intriguing topic in contemporary mathematics.