The Erdos-Borwein constant E is the sum 1/(2¹−1) + 1/(2²−1) + 1/(2³−1) + ⋯ = 1/1 + 1/3 + 1/7 + 1/15 + 1/31 + ⋯ The denominators are the Mersenne numbers 2ⁿ − 1. Paul Erdos proved in 1948 that E is irrational, using only elementary properties of binary representations.
The partial sums converge quickly to E ≈ 1.6066951524. The denominators 2^n−1 grow geometrically, making convergence much faster than the Basel problem.
The series converges geometrically fast: each term is roughly half the previous one (since 2ⁿ − 1 ≈ 2ⁿ for large n). After just 20 terms the sum is accurate to 6 decimal places. The equivalence E = Σ d(n)/2ⁿ (where d(n) counts odd divisors of n) links it to divisibility theory.
Whether E is transcendental is open. What makes Erdos's irrationality proof memorable is its economy: he used the fact that the binary representations of the denominators 1, 3, 7, 15, 31… (which are 1, 11, 111, 1111, 11111 in binary) have a special structure that prevents the sum from being rational. The value: 1.60669515245214159769492939967985…
Each denominator 2^n - 1 is roughly twice the previous. Sum converges to E ~1.6066951524.
The Erdos-Borwein constant E = 1/1 + 1/3 + 1/7 + 1/15 + ... ≈ 1.60669. Paul Erdos proved it irrational in 1948 using binary properties of the denominators 2^n - 1. It equals the sum of d(n)/2^n where d(n) counts odd divisors of n. The series converges rapidly: each term is roughly half the previous. Whether it is transcendental is unknown. Value: 1.60669515245214159769492939967985...