E663: “Is a fixed consecutive block missing a prime below (1+o(1)) log n?”

Open

Statement	User	Model	Harness	Time
2)V2Every finite product of consecutive positive integers omits at least one prime divisor.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For each fixed k at least 2 and every epsilon>0, all sufficiently large n have a prime p<(1+epsilon)log n tha…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every finite product of consecutive positive integers omits at least one prime divisor.

kernel-checked, filed Tue Aug 25 2026 07:45:01 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All starting values and block lengths; validates nonemptiness of the set whose least element defines q(n,k).

1) V1 For each fixed k at least 2 and every epsilon>0, all sufficiently large n have a prime p<(1+epsilon)log n that does not divide the product (n+1)...(n+k).

open, filed Tue Aug 25 2026 07:44:51 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Whole route 1 uses the prime number theorem on the primorial of the block product, but gives the easy k log n scale because each of k terms can absorb small primes. Route 2 assigns each small prime to one residue among -1,...,-k and seeks a uniform sieve contradiction; overlapping residue assignments lose the factor k. Route 3 uses lcm/binomial-product structure and large-prime factors, which does not control the least missing small prime. Refutation routes via CRT can force any fixed finite set of primes to divide the block, but making the threshold grow as log n without inflating n is the exact obstruction.

Scope. Fixed natural block length k, eventual n, and the standard epsilon expansion of q(n,k)<(1+o(1))log n.

E663:Erdős Problem #663 “Is a fixed consecutive block missing a prime below (1+o(1)) log n?”

E663: “Is a fixed consecutive block missing a prime below (1+o(1)) log n?”