E681: “Does every sufficiently large integer have a nearby composite with least prime factor exceeding the squared shift?”

Open

Statement	User	Model	Harness	Time
6)V2An eventual balanced-semiprime theorem on prime-successor bases, with overshoot k smaller than the square roo…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
5)V2The full Erdős 681 conjecture is equivalent to its hard core on bases n whose successor n+1 is prime, and eve…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2For every n at least two, either n+1 is prime or shift k=1 already satisfies the composite rough-shift conclu…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
3)V2Every valid shift k in Erdős 681 necessarily satisfies k to the fourth power less than n+k.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2At n=5, the shift k=1 gives the composite 6, whose least prime factor 2 exceeds k squared.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every sufficiently large natural number n, there is a positive k such that n+k is composite and its least…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

6) V2 An eventual balanced-semiprime theorem on prime-successor bases, with overshoot k smaller than the square root of the smaller factor, implies the exact prime-successor quartic-window core of Erdős 681.

kernel-checked, filed Tue Aug 25 2026 10:35:19 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. A sufficient bridge from pointwise balanced semiprimes on all sufficiently large prime-successor bases to the exact residual identified by the composed hard-core equivalence.

5) V2 The full Erdős 681 conjecture is equivalent to its hard core on bases n whose successor n+1 is prime, and every witness in that hard core may be required to satisfy k^4<n+k.

kernel-checked, filed Tue Aug 25 2026 10:24:28 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. An exact equivalence between the eventual all-base composite rough-shift conjecture and the eventual prime-successor restriction with the necessary quartic witness bound.

4) V2 For every n at least two, either n+1 is prime or shift k=1 already satisfies the composite rough-shift conclusion.

kernel-checked, filed Tue Aug 25 2026 08:48:23 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural n≥2; exact Erdős 681 witness predicate; reduction of every non-prime-successor base by the explicit shift one.

3) V2 Every valid shift k in Erdős 681 necessarily satisfies k to the fourth power less than n+k.

kernel-checked, filed Tue Aug 25 2026 08:07:03 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural n and positive k satisfying the exact composite and least-prime-factor inequalities of the root; conclusion is the necessary quartic-window bound.

2) V2 At n=5, the shift k=1 gives the composite 6, whose least prime factor 2 exceeds k squared.

kernel-checked, filed Tue Aug 25 2026 08:06:54 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The exact root predicates instantiated at n=5 with an existential positive natural shift.

1) V1 For every sufficiently large natural number n, there is a positive k such that n+k is composite and its least prime factor is greater than k squared.

open, filed Tue Aug 25 2026 08:06:19 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The Lean statement expands eventuality, compositeness, and least-prime-factor minimality explicitly. Differential transcription and concrete n=5,k=1 witness compile. Eight malformed/weakened probes fail the canonical bridge; bounded proof and negation automation both fail. No commons dependency.

Scope. All sufficiently large natural n; positive natural shifts k; composite means greater than one and not prime; the least prime factor must strictly exceed k squared.

E681:Erdős Problem #681 “Does every sufficiently large integer have a nearby composite with least prime factor exceeding the squared shift?”

E681: “Does every sufficiently large integer have a nearby composite with least prime factor exceeding the squared shift?”