The problem
Is it true that for every infinite sequence f: ℕ → {−1,+1} and every C > 0, there exist d, k ≥ 1 with |∑_{j≤k} f(jd)| > C? (Discrepancy along some homogeneous progression is unbounded.)
The Erdős Discrepancy Problem
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History & significance
Posed by Erdős circa 1932; he offered $1000 for it, calling discrepancies of multiplicative functions a matter of principle. Partial results: Roth, Sárközy (multiplicative case), Beck (two-dimensional). In 2014 Konev and Lisitsa's SAT experiments found a length-1160 sequence of discrepancy 2 and suggested discrepancy-3 sequences fail near 127,645 — computational hints begging for theory.
The resolution (human proof)
Proven September 2015 by Terence Tao, weeks after Polymath revived the problem with those SAT clues: a six-page argument proves discrepancy grows faster than any constant — in fact logarithmically — via an entropy-decrement argument transferring multiplicative-function estimates to arbitrary signs. A model case of computer-experiment-guided human insight: the machines located the cliff; a Fields medallist explained why no one survives falling off it.