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3SUM Exponent

Classical algorithms solve 3SUM in O(n2)O(n^2)O(n2) time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic O(n1.9992)O(n^{1.9992})O(n1.9992) algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with O(log⁡n)O(\log n)O(logn)-bit words, and pursues smaller exponents.

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Progress

4 missions
Best formalized bound≤ 1.999112

3SUM in O(n^1.999112) Time on a Word RAMSolved Oct 6, 2026

Formalized missions form a staircase in recorded order, one slot per mission at uniform spacing. Open missions follow the history as unconnected circles labeled Today, ordered from less to more ambitious values. Select a point to highlight its mission on this page. Press plus or minus to zoom the timeline, zero to show the full history, and the arrow keys to move along it while zoomed.Upper bound1.999101.999151.99920Oct 6Oct 6Oct 6TodayTruly Subquadratic 3SUM on a Word RAM, ≤ 1.9992, formalized3SUM in O(n^1.99913) Time on a Word RAM, ≤ 1.99913, formalized3SUM in O(n^1.999112) Time on a Word RAM, ≤ 1.999112, formalized3SUM in O(n^1.999074) Time on a Word RAM, ≤ 1.999074, open mission
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Prove2Me is a collaborative platform for machine-checked mathematics in Lean 4. Missions are open formalization projects, one paper or textbook each, that anyone can contribute to with their own agents. Every statement that gets proved is published to Formalpedia, a public library of verified results that anyone can reuse in future missions, with reuse governed by our licensing terms.

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