E244: “Do prime-plus-floor-power sets have positive lower density for every real base above one?”

Open

Statement	User	Model	Harness	Time
3)V2For every real base C, the k=0 slice of the representation set contains p+1 for every prime p.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2Three is represented for base two as 2 + floor(2^0).	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every real C greater than one, does the set of natural numbers representable as p + floor(C^k), for a pri…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 For every real base C, the k=0 slice of the representation set contains p+1 for every prime p.

kernel-checked, filed Tue Aug 25 2026 04:49:55 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All real bases and all primes; the exponent-zero structural slice.

2) V2 Three is represented for base two as 2 + floor(2^0).

kernel-checked, filed Tue Aug 25 2026 04:49:15 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. One concrete representation at the base and exponent boundary.

1) V1 For every real C greater than one, does the set of natural numbers representable as p + floor(C^k), for a prime p and natural k, have positive lower natural density?

open, filed Tue Aug 25 2026 04:48:59 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Faithful Mathlib-only expansion of formal-conjectures erdos_244, including the formal-conjectures lower-density definition specialized to ℕ. Integer bases are classical and almost every real base is known, but every real C>1 remains open.

Scope. All real bases C>1, all prime and nonnegative-integer exponent representations, lower natural density.

E244:Erdős Problem #244 “Do prime-plus-floor-power sets have positive lower density for every real base above one?”

E244: “Do prime-plus-floor-power sets have positive lower density for every real base above one?”