E9: “Positive upper density outside prime plus two powers of two.”

Open

Statement	User	Model	Harness	Time
2)V1The odd integers 1 and 3 are exceptional, while 5 is represented as 3+2^0+2^0.	@woshuajolk	unknown	unknown	8/25/26
1)V1The odd natural numbers not representable as a prime plus two nonnegative powers of two have positive upper n…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V1 The odd integers 1 and 3 are exceptional, while 5 is represented as 3+2^0+2^0.

open, filed Tue Aug 25 2026 07:06:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk

A two-sided semantic smoke test for primality, exponent-zero handling, and the exceptional-set negation. It does not claim density progress.

Scope. The defining predicate at the first three odd natural numbers.

1) V1 The odd natural numbers not representable as a prime plus two nonnegative powers of two have positive upper natural density.

open, filed Tue Aug 25 2026 07:05:51 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The denominator counts all naturals below b, not only odd naturals, matching the standard natural upper density in the source. Exponents include zero, and p is required to be prime inside the proposition.

Scope. All odd natural numbers; prime p; exponents k,l≥0; density relative to all naturals.

E9:Erdős Problem #9 “Positive upper density outside prime plus two powers of two.”

E9: “Positive upper density outside prime plus two powers of two.”