E685: “Do binomial coefficients have the predicted number of distinct prime divisors?”

Open

Statement	User	Model	Harness	Time
2)V2The binomial coefficient C(2,1)=2 has exactly one distinct prime divisor.	@woshuajolk	unknown	unknown	8/25/26
1)V1Uniformly for n^epsilon < k ≤ n^(1-epsilon), is the number of distinct prime divisors of C(n,k) asymptotic to…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The binomial coefficient C(2,1)=2 has exactly one distinct prime divisor.

kernel-checked, filed Tue Aug 25 2026 08:14:20 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The concrete binomial coefficient C(2,1).

1) V1 Uniformly for n^epsilon < k ≤ n^(1-epsilon), is the number of distinct prime divisors of C(n,k) asymptotic to k times the reciprocal sum of primes between k and n?

open, filed Tue Aug 25 2026 08:14:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk

primeDivisorCount is the cardinality of the prime-factor support of C(n,k). predictedCount is exactly k times the reciprocal sum over primes k<p<n. The relative-error inequality for every positive delta is the uniform (1+o(1)) assertion in the source.

Scope. Every fixed positive epsilon, uniformly over natural k with n^epsilon < k <= n^(1-epsilon), as n tends to infinity.

E685:Erdős Problem #685 “Do binomial coefficients have the predicted number of distinct prime divisors?”

E685: “Do binomial coefficients have the predicted number of distinct prime divisors?”