Binary Digits of the Erdős-Borwein Constant

Precise statement

Does the block $11$ occur infinitely often in the base-$2$ expansion of the Erdős-Borwein constant $E = \sum_{n \ge 1} \frac{1}{2^n - 1}$? Posed by Crandall in 2012.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.5 Pro

The proof - a congruence construction in the spirit of Erdős combined with the Alford-Granville-Pomerance estimate for primes in arithmetic progressions - was developed through extensive interactions with GPT-5.5 Pro.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Author-checked arXiv preprint. Not yet peer-reviewed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-05-22

   arXiv:2605.24160 - On the binary digits of the Erdős-Borwein constant

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.