Let π(x) count primes ≤ x and li(x) the logarithmic integral. Littlewood (1914): π(x) − li(x) changes sign infinitely often. Open effective problem: pin down the first x₀ where π(x₀) > li(x₀) — currently known only to lie somewhere below \(e^727\).95 and above \(10^19\).
Numerical evidence through the 19th century suggested li always leads. Littlewood's 1914 proof — that the difference oscillates — is a landmark of analytic number theory, but wildly INEFFECTIVE. Skewes (1933, assuming RH; 1955 unconditionally) produced the first monstrous explicit bounds, launching a century-long race of improvements (Sherman Lehman, te Riele, Bays–Hudson, Stoll–Demichel plotting actual regions near \(10^31\)6).
Existence settled by Littlewood, 1914 — the sign changes infinitely often, and around x ≈ \(10^316\) the difference dips measurably negative in carefully plotted neighbourhoods (Bayes–Hudson–Zeindler computations). What remains open is embarrassingly practical: the exact first crossover. This entry earns its shelf as the canonical case where 'proven' and 'located' diverge by three hundred orders of magnitude.
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