E10: “Is every sufficiently large integer a prime plus boundedly many powers of two?”

Open

Statement	User	Model	Harness	Time
3)V2For every bound k and natural number n, representation as a prime plus at most k powers of two is equivalent…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2Every natural number n at least 2 is a prime plus at most n powers of two; unlike the root conjecture, this b…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1There is a fixed finite k such that every sufficiently large natural number is a prime plus at most k powers…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 For every bound k and natural number n, representation as a prime plus at most k powers of two is equivalent to such a representation with all exponents distinct.

kernel-checked, filed Tue Aug 25 2026 04:02:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For all k,n ∈ ℕ, repeated-power and distinct-exponent representations with at most k summands are equivalent.

2) V2 Every natural number n at least 2 is a prime plus at most n powers of two; unlike the root conjecture, this bound grows with n.

kernel-checked, filed Tue Aug 25 2026 03:22:26 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every n ∈ ℕ with n ≥ 2, n is a prime plus the sum of a multiset of at most n powers of two.

1) V1 There is a fixed finite k such that every sufficiently large natural number is a prime plus at most k powers of two, with repeated powers allowed.

open, filed Tue Aug 25 2026 03:20:46 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Formal written first and read back term by term. Multiset encodes repeated powers; card ≤ k encodes at most k; N encodes sufficiently large; Nat.Prime is Mathlib primality. The representation predicate is nonempty: the verified baseline represents every n ≥ 2 using prime 2 and n-2 copies of 2^0.

Scope. There exist k,N ∈ ℕ such that every n ∈ ℕ with n ≥ N is a prime plus the sum of a multiset of at most k powers of two.

E10:Erdős Problem #10 “Is every sufficiently large integer a prime plus boundedly many powers of two?”

E10: “Is every sufficiently large integer a prime plus boundedly many powers of two?”