E236: “Is the number of prime-plus-power-of-two representations sublogarithmic?”

Open

Statement	User	Model	Harness	Time
2)V2For every natural n, the number of exponents k considered in the prime-plus-power-of-two representation count…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Let f(n) count the exponents k ≥ 0 for which n - 2^k is prime.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 For every natural n, the number of exponents k considered in the prime-plus-power-of-two representation count is at most floor(log₂ n) + 1.

kernel-checked, filed Tue Aug 25 2026 04:18:00 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The exact representation-count definition used in the root statement; a pointwise elementary upper bound for every natural n.

1) V1 Let f(n) count the exponents k ≥ 0 for which n - 2^k is prime.

open, filed Tue Aug 25 2026 04:16:00 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Prove that f(n) = o(log n).

Faithfully mirrors the formal-conjectures definition. Finite witnesses 3, 4, 7, and 15 establish non-vacuity; an independent encoding is definitionally equal; ten content-free bridges are rejected. Full routes explored: the elementary count gives only O(log n); fixed congruence sieves leave positive exponent density; growing sieves require the unresolved uniform analytic estimate; existing representability-density results do not imply a pointwise bound. The 1950 Ω(log log n) lower bound is compatible with the conjecture.

Scope. Natural n; exponents 0 through floor(log₂ n), counted once each; primality in ℕ; little-o along n tending to infinity.

E236:Erdős Problem #236 “Is the number of prime-plus-power-of-two representations sublogarithmic?”

E236: “Is the number of prime-plus-power-of-two representations sublogarithmic?”