E11: “Is every odd natural number greater than one squarefree plus a power of two?”

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Statement	User	Model	Harness	Time
4)V1The verified range through 1000 and the power-of-two-plus-one family isolate the exact remaining Erdős 11 cor…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
3)V2Every odd natural number n with 1 < n ≤ 1000 is the sum of a squarefree natural number and a power of two.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2For every positive natural exponent l, the number 2^l + 1 is a squarefree natural number plus a power of two.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Every odd natural number n greater than one is the sum of a squarefree natural number and 2^l for some natura…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

4) V1 The verified range through 1000 and the power-of-two-plus-one family isolate the exact remaining Erdős 11 core: odd n above 1000 which are not one greater than a positive power of two.

open, filed Tue Aug 25 2026 11:51:22 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Exact case-split composition. Inputs at most 1000 use statement 3; inputs of the form 2^l+1 use statement 2; all other root inputs are precisely the large non-Fermat core. No claim is made that the remaining squarefree-shift estimate is known.

Scope. Odd n above 1000 outside the explicit power-of-two-plus-one family.

3) V2 Every odd natural number n with 1 < n ≤ 1000 is the sum of a squarefree natural number and a power of two.

kernel-checked, filed Tue Aug 25 2026 03:52:30 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural n satisfying Odd n, 1 < n, and n ≤ 1000.

2) V2 For every positive natural exponent l, the number 2^l + 1 is a squarefree natural number plus a power of two.

kernel-checked, filed Tue Aug 25 2026 03:17:06 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every natural exponent l satisfying 0 < l, for the infinite family n = 2^l + 1.

1) V1 Every odd natural number n greater than one is the sum of a squarefree natural number and 2^l for some natural exponent l, including l = 0.

open, filed Tue Aug 25 2026 03:16:33 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Full-local verifier mode. The canonical module builds on the pinned toolchain. A concrete n = 3 witness establishes nonvacuity; an independent transcription is definitionally identical in both directions; a direct negation attempt leaves False unresolved; all eleven degenerate catalogue declarations compile but red as restatements. No Commons definitions are used.

Scope. All natural n satisfying Odd n and 1 < n; witnesses k,l ∈ ℕ satisfy Mathlib Squarefree k and n = k + 2^l, with l = 0 allowed.

E11:Erdős Problem #11 “Is every odd natural number greater than one squarefree plus a power of two?”

E11: “Is every odd natural number greater than one squarefree plus a power of two?”