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On the binary digits of the Erd\H{o}s-Borwein constant

This paper provides an affirmative proof that the binary string "11" occurs infinitely often in the base-2 expansion of the Erdős-Borwein constant, resolving a 2012 open problem posed by Crandall through a novel combination of Erdős-style congruence constructions and prime counting estimates, with significant development aided by AI.

Original authors: John M. Campbell

Published 2026-05-26

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: John M. Campbell

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical number, let's call it E. This number is special because mathematicians have known for a long time that it's "irrational," meaning its decimal (or in this case, binary) expansion goes on forever without ever repeating a pattern.

The number E is built by adding up a specific list of fractions:
$E = \frac{1}{2^1-1} + \frac{1}{2^2-1} + \frac{1}{2^3-1} + \dots$

When you write E out in binary (using only 0s and 1s), it looks like this:
1.1001101101010000010111111...

The Big Question

In 2012, a mathematician named Richard Crandall asked a simple but tricky question: Does the pattern "11" (two ones in a row) appear infinitely many times in this endless string of binary digits?

It's like asking: If you keep flipping a coin forever, will you eventually see "Heads-Heads" an infinite number of times? For this specific number E, nobody knew the answer for over a decade.

The Solution

The author of this paper, John Campbell, says: "Yes, it does."

He didn't just guess; he built a mathematical proof to show that no matter how far you go into the number E, you will always find more "11" patterns waiting for you.

How Did He Do It? (The Analogy)

To prove this, the author used a clever construction, kind of like building a custom lock and key system.

The "Divisor" Counting Game:
The number E is secretly connected to how many "divisors" (factors) numbers have. For example, the number 6 has four divisors (1, 2, 3, 6). The author needed to find specific numbers where the count of divisors behaves in a very specific way.
The "Chinese Remainder" Lock:
Imagine you have a giant safe with many different locks. Each lock only opens if you turn the dial to a specific number. The author used a famous mathematical tool called the Chinese Remainder Theorem. Think of this as a master key that can find a single number that satisfies all these different locks at the same time.

He designed a system of locks so that when he found the "key" number (let's call it $n$ ), the number $n+2$ would have exactly 6 divisors, and other nearby numbers would have a huge number of divisors.
The "Prime" Hunt:
To make sure this "key" number actually exists and is big enough, he had to find a lot of special prime numbers (numbers divisible only by 1 and themselves). He used a map of prime numbers (based on work by Alford, Granville, and Pomerance) to guarantee there were enough "prime ingredients" to build his lock system.
The "Tail" Problem:
When calculating the value of E, the author had to worry about the "tail"—the infinite sum of tiny fractions at the very end. He had to prove that this tail was so small it wouldn't mess up the pattern he was looking for. He showed that the "noise" at the end of the calculation was too quiet to hide the "11" pattern.

The Result

By combining these tools, the author proved that he could find a starting point $n$ where:

The $n$ -th digit of E is a 1.
The $(n+1)$ -th digit of E is also a 1.

Because he could make this starting point $n$ as large as he wanted (by making his "locks" bigger and finding bigger primes), he proved that the "11" pattern doesn't just happen once or twice; it happens infinitely often.

A Note on AI

The paper includes a unique admission: the author developed this complex proof through extensive collaboration with an AI named GPT-5.5 Pro. However, the author emphasizes that the AI only offered suggestions, which he then heavily revised, corrected, and verified. He takes full responsibility for the final math.

Summary

In short: The paper solves a 12-year-old mystery about a famous number. It proves that if you look deep enough into the binary code of the Erdős–Borwein constant, you will never run out of "11" pairs. The proof is a masterclass in using prime numbers and modular arithmetic to force a specific pattern to appear.

Technical Summary: On the binary digits of the Erdős–Borwein constant

Problem Statement
The paper addresses an open problem regarding the base-2 (binary) expansion of the Erdős–Borwein constant, $E$ , defined by the Lambert series:
$E = \sum_{n=1}^{\infty} \frac{1}{2^n - 1} = \sum_{n=1}^{\infty} \frac{d(n)}{2^n}$
where $d(n)$ is the number of positive divisors of $n$ . While Erdős (1948) proved that $E$ is irrational, the specific distribution of its binary digits remained partially unresolved. In 2012, Crandall investigated the parity of specific digits of $E$ and posed the question of whether the binary string "11" (two consecutive ones) occurs infinitely often in the expansion of $E$ . This problem was subsequently highlighted by Shallit. The paper aims to provide a full proof that the string "11" appears infinitely often in the base-2 expansion of $E$ .

Methodology
The proof employs a constructive approach combining number-theoretic tools, specifically the Chinese Remainder Theorem (CRT) and estimates on the distribution of primes in arithmetic progressions. The methodology proceeds through the following steps:

Congruence Construction: The authors construct a sequence of integers $n$ (dependent on a large parameter $X$ ) such that the divisor function $d(n+2)$ takes a specific value ( $d(n+2)=6$ ) and the tail of the series expansion of $E$ starting from index $n$ is sufficiently small.
Application of the CRT: A system of simultaneous congruences is solved to define a residue class $r$ modulo a carefully constructed modulus $A$ . This modulus $A$ is formed by products of distinct primes chosen from a specific interval $(L, 2L)$ , where $L \approx (\log X)^2$ .
Prime Distribution (AGP Lemma): To ensure the existence of suitable primes for the modulus construction, the paper utilizes a lemma by Alford, Granville, and Pomerance (AGP). This lemma provides a lower bound for the counting function $\pi(N; d, a)$ (primes $\le N$ congruent to $a \pmod d$ ), ensuring that primes exist in the required arithmetic progressions that avoid a specific "exceptional set" of moduli.
Divisor Function Bounds: The proof relies on bounding the sum of divisor functions over arithmetic progressions (Lemma 2) and estimating the "tail" of the series $\sum d(n+\ell)2^{-\ell}$ (Lemma 3). These bounds ensure that the contribution of terms beyond a certain point does not disrupt the integer properties of the partial sums.
Pigeonhole Argument: By comparing the number of integers $m$ for which $d(n_m+2)=6$ against the number of integers for which the tail sum is "large," the authors demonstrate the existence of at least one integer $n$ where the tail is small enough to force the binary digits at positions $n$ and $n+1$ to be 1.

Key Contributions and Results

Resolution of Crandall's Problem: The paper provides a rigorous proof that the string "11" occurs infinitely often in the binary expansion of the Erdős–Borwein constant.
Construction of Specific Integers: The authors explicitly construct a sequence of integers $n$ such that the $n$ -th and $(n+1)$ -th binary digits of $E$ are both 1. This is achieved by showing that for sufficiently large $X$ , there exists an $n$ where $2^{n-1}E$ has a fractional part in the interval $[3/4, 1)$ .
Integration of Advanced Number Theory: The work synthesizes Erdős's original irrationality proof techniques (congruence constructions involving strings of zeros) with modern estimates on primes in arithmetic progressions (AGP lemma) to tackle a digit-distribution problem.
Tail Estimates: The paper establishes specific bounds on the tail of the Lambert series expansion for $E$ , demonstrating that the "noise" from higher-order terms can be controlled to reveal specific digit patterns.

Significance and Claims
The paper claims to solve Crandall's 2012 open problem in the affirmative. The significance lies in advancing the understanding of the arithmetic properties of the Erdős–Borwein constant, specifically its normality properties. While the paper does not prove that $E$ is 2-normal (i.e., that every finite binary string appears with the expected frequency), it establishes a necessary condition for normality by proving the infinite occurrence of a specific non-trivial string ("11").

The authors note that their argument was developed through extensive interactions with GPT-5.5 Pro, though all AI-generated suggestions were substantially revised, corrected, and independently verified by the author. The paper concludes by suggesting that the next natural step is to investigate whether every binary string appears infinitely often in the expansion of $E$ .