#122: does n+f(n) pile up on the same values, for (almost) every slow-growing f?
The target (erdosproblems.com/122, OPEN, 0 claimed proofs — three people currently working on it). For which number-theoretic f is it true that for every F with F(n)/f(n)→0 almost everywhere, there are infinitely many x where the count of n with n+f(n)∈(x,x+F(x)), divided by F(x), goes to ∞? Erdős, Pomerance and Sárközy proved a concrete clustering statement for f=ω(n) (number of distinct prime factors) and reported — without publishing details — that the same works for f=τ(n) (the divisor function), and probably fails for f=φ(n) or σ(n).
Three real, separable findings, none of them the full theorem. (1) Maynard's 2016 prime-clustering theorem really does give an explicit, verifiable construction of arbitrarily large exact fibers of n+τ(n) — a CRT-forced system of affine forms where forcing divisors with prescribed τ-values and then hitting several simultaneous primes creates exact collisions, checked algebraically end to end. (2) The actual published EPS mechanism for ω(n) isn't "shifts don't change ω(n) much" (a plausible-sounding wrong guess) — it's a CRT construction that programs a forced additive contribution to ω(n+i) that exactly cancels the offset i, with a controlled statistical residual doing the rest. (3) That mechanism breaks for τ(n) for a precise algebraic reason: ω is additive over coprime factors, τ is multiplicative — so the same offset-cancellation trick that works by addition for ω needs multiplication for τ, and the natural probabilistic estimate (normal order of log τ) becomes useless once you exponentiate back to τ itself. EPS's own printed remark that τ works "with a little more difficulty" is real, but the details were never published, and a plausible reconstruction of what they might have meant runs into three concrete new gaps, honestly named rather than papered over.
We checked the checkable parts ourselves. Built a small real instance of the exact-collision construction from scratch — k=(4,6,8), primes (5,7,11), d_i=p_i^(k_i/2−1)=(5,49,1331), verified τ(d_i) really equals k_i/2 exactly, ran CRT for b and D, then searched for t where multiple L_i(t) are simultaneously prime. The very first hit, at t=0: y=81199, with n=81193 and n=81191 both independently satisfying n+τ(n)=81199 — a genuine, verified double collision, not asymptotic, not simulated. Also directly tested the paper's general density constraint: sieved τ(n) for n up to 5,000,000, confirmed Σ_{y≤X} r_τ(y) = 4,999,983 (essentially X, as claimed) and that the density of {y: r_τ(y)≥M} stays comfortably under 1/M at every M we checked (found real fibers up to size 7 in that range). Every citation (Maynard, Compositio Math 152 (2016), 1517–1554; Erdős–Pomerance–Sárközy I, J. Number Theory 21 (1985); EPS IV, Ramanujan J. 1 (1997)) verified directly against Crossref, exact match on title/venue/volume/pages.
A real source-hygiene catch, resolved. The source chat flagged that older archived versions of this problem's statement may have had the inequality direction reversed (f/F→0 vs. F/f→0) and couldn't access the live tracker to check. We could: the page as it stands today uses F(n)/f(n)→0, matching what the analysis assumed — so that specific worry is resolved, though the original 1997 Erdős source text itself still hasn't been independently re-read by us or the chat.
What's still open. No proof of the general arbitrary-F statement, not even reconstructed for f=ω. A fixed Maynard tuple provably can never give positive density of collision centers (its centers sit inside a finite union of prime-value sequences, density zero) — real news, but it also means the eventual proof needs a genuinely new averaged construction over a growing family of systems, not a stronger single application of Maynard. The named next-step lemmas (a divisor-packet efficiency lemma, a uniform exact-ω residual lemma) are plausible but unproven.