Binary Structure of Even Perfect Numbers: p Ones Followed by p−1 Zeros — E8 Intelligence Research

FINDING: Perfect numbers in binary are a string of p ones followed by (p−1) zeros, a direct consequence of Euclid–Euler form for even perfect numbers. | MATH: Even perfect number \( N = 2^{p-1}(2^p - 1) \) where \( 2^p - 1 \) is Mersenne prime. Binary representation: \( N = (111\ldots 1)_2 \) with \( p \) ones, followed by \( (p-1) \) zeros. Proof: \( 2^p - 1 = (111\ldots 1)_2 \) (p ones); multiplying by \( 2^{p-1} \) shifts left by \( p-1 \) bits, appending zeros. Example: \( p=3 \), \( N=28 = 11100_2 \) (3 ones, 2 zeros). | CONNECTION: The binary pattern \( 1^p 0^{p-1} \) is a geometric sequence in base 2: sum of \( p \) ones = \( 2^p - 1 \), then shift. This mirrors base-10 repunits (e.g., 999…9) and relates to the ratio \( 2^p/(2^p-1) \to 1 \) as \( p \to \infty \), but for finite \( p \), the ratio \( N / 2^{2p-1} \approx 1/2 \) — a self-similar halving symmetry. The structure \( 1^p 0^{p-1} \) is a binary palindrome-like pattern with a single transition, reflecting a one-dimensio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com