Density of powers of two plus almost-primes
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Can almost all positive integers be expressed as 2ᵏ+n with n having a bounded number of prime factors? The target allows the factor bound r to depend on a requested lower-density loss ε>0.
Why it matters and what counts as progress
Why it matters: Density approaching one strengthens sparse representation results while allowing the almost-prime complexity to grow with the requested accuracy.
What counts as progress: Prove lower density at least 1−ε for some r(ε), improve r or obtain a natural-density conclusion with the assumptions stated separately.
How the problem developed
- 1934 · Theorem [1]
Nikolai Romanoff Proved that numbers of the form 2^k+p, with p prime, have positive lower density; this is weaker than density arbitrarily close to one with a bounded number of prime factors.
- 1985 · Open problem [1]
Paul Erdős Asked whether sums of a power of two and an integer with a bounded number of prime factors have density arbitrarily close to one.
- 5 Feb 2026 · Reported claim [1]
GPT-5.2 Pro, attributed to Liam Price The problem thread reports an affirmative solution attributed to Liam Price with GPT-5.2 Pro; discussion notes a separate quantitative argument by Ben Green and Terence Tao and records Tao’s check of Price’s proof.
AI contribution
GPT-5.2 Pro sieve proof gives near-one lower density for Erdős #851
For every ε>0, there is an r such that integers 2^k+n with n having at most r prime factors have lower density at least 1−ε. The claim neither establishes a natural density nor fixes one r for all ε.
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GPT-5.2 Pro sieve proof gives near-one lower density for Erdős #851