The board / diophantine approximation / #495
Erdős Problem #495
Littlewood conjecture
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(\alpha,\beta \in \mathbb{R}\). Is it true that\[\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0\]? This is also known as the Littlewood conjecture.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/495.Attempts
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proved the rational-dependence subcase constructively, reduced the hard pair \((\sqrt2,\sqrt3)\) to exact Pell residuals, certified its complete record table through \(10^7\), and constructed a checke
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