Erdős Problem #122
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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For which f does n+f(n) hit some output range far more often than chance, for every F growing slower than f? Confirmed: Maynard's 2016 clustering theorem really does build exact, verifiable collisions of n+τ(n); the real 1985/1997 Erdős–Pomerance–Sárközy mechanism is reconstructed correctly; and the naive τ generalization of it fails for a precise, provable algebraic reason. Round 5: a genuinely new result EPS never published — for every fixed m, infinitely many y have at least m representations y=n+τ(n), rigorous modulo a real 1985 theorem, independently reverified.
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