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Limits RegistryTerence Tao proved the Erdős Discrepancy Problem in 2015, showing that the discrepancy of any infinite ±1 sequence along arithmetic progressions is unbounded, resolving a conjecture Paul Erdős posed around 1932. The proof built on the Polymath5 collaborative project and, notably, on a computer-generated 13-gigabyte SAT-solver proof by Boris Konev and Alexei Lisitsa (2014) that had already established the discrepancy exceeds 2, combined with new analytic number theory results connected to a partial resolution of the Elliott conjecture.
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Terence Tao proved the Erdős Discrepancy Problem in 2015, showing that the discrepancy of any infinite ±1 sequence along arithmetic progressions is unbounded, resolving a conjecture Paul Erdős posed around 1932. The proof built on the Polymath5 collaborative project and, notably, on a computer-generated 13-gigabyte SAT-solver proof by Boris Konev and Alexei Lisitsa (2014) that had already established the discrepancy exceeds 2, combined with new analytic number theory results connected to a partial resolution of the Elliott conjecture.
For any infinite sequence of +1s and -1s and any constant C, there exist integers d and k such that the sum of every d-th term up to the k-th multiple of d exceeds C in absolute value — i.e. the discrepancy of any ±1 sequence along homogeneous arithmetic progressions is unbounded.
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Limits Registry. LR-ERDOS-DISCREPANCY-PROBLEM. Proof of the Erdős Discrepancy Problem. 2026.