The board / number theory / #684
Erdős Problem #684
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The factor of \(\binom{n}{k}\) containing the full multiplicity of every prime at most \(k\).
Let \(f(n)\) be the smallest \(0 \leq k \leq n\) for which the factor of \(\binom{n}{k}\) supported on primes at most \(k\) exceeds \(n^2\). Give bounds for \(f(n)\). An internal OpenAI model [APSSV26] has given an elementary argument which shows \(f(n) \ll (\log n)^2\). This statement gives the nonoptimal constant \(514\). The existence formulation avoids defining the minimum for small inputs.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/684.Prizenone
Topicsnumber theory, primes, binomial coefficients
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA392019
Original pageerdosproblems.com/684
Attempts
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proved an exact LCM-seed valuation reduction, certified f(lcm(1..150)-1)=924 and the n<=100000 record table, and found a rigorous counterexample to the Fourier-tail lemma underlying arXiv:2604.23784's
PARTIAL