New Mersenne Prime Found; Unverified Theorem Links Perfect Numbers to 6-Power Primes — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Perfect numbers remain tied to Mersenne primes via Euclid–Euler; a new Mersenne prime (2^136279841 − 1) was discovered, and a claimed theorem on primes of form 6^(m+1)·N − 1 exists but is unverified. | MATH: Euclid–Euler: even perfect number N = 2^(p−1)·(2^p − 1) iff 2^p − 1 is prime (Mersenne). New prime: p = 136279841, digits = 41,024,320. Claimed: P = 6^(m+1)·N − 1 prime for 1 < N ≤ 13, N ≠ 8, N ≠ i^(m+1) mod (6i+1), m odd positive; extension for N > 13 conditional. | CONNECTION: Mersenne primes 2^p − 1 relate to binary repunits (all 1s in base 2) — a lattice of dimension p. Perfect numbers are triangular: N = 2^(p−1)(2^p − 1) = T_(2^p − 1) (triangular number), linking to hexagonal numbers (T_n = n(n+1)/2; hexagonal = n(2n−1)). The ratio of perfect number to its Mersenne prime: N/(2^p − 1) = 2^(p−1) — a power of 2, not a golden ratio. However, the sum of reciprocals of divisors of a perfect number = 2 (harmonic mean of divisors is an integer), echoing the 2:1 octave ratio i
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026mersenne,
title = {{New Mersenne Prime Found; Unverified Theorem Links Perfect Numbers to 6-Power Primes — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490940},
url = {https://zenodo.org/records/23490940},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - New Mersenne Prime Found; Unverified Theorem Links Perfect Numbers to 6-Power Primes — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490940
UR - https://zenodo.org/records/23490940
AB - FINDING: Perfect numbers remain tied to Mersenne primes via Euclid–Euler; a new Mersenne prime (2^136279841 − 1) was discovered, and a claimed theorem on primes of form 6^(m+1)·N − 1 exists but is unverified. | MATH: Euclid–Euler: even perfect number N = 2^(p−1)·(2^p − 1) iff 2^p − 1 is prime (Mersenne). New prime: p = 136279841, digits = 41,024,320. Claimed: P = 6^(m+1)·N − 1 prime for 1 < N ≤ 13, N ≠ 8, N ≠ i^(m+1) mod (6i+1), m odd positive; extension for N > 13 conditional. | CONNECTION: Mersenne primes 2^p − 1 relate to binary repunits (all 1s in base 2) — a lattice of dimension p. Perfect numbers are triangular: N = 2^(p−1)(2^p − 1) = T_(2^p − 1) (triangular number), linking to hexagonal numbers (T_n = n(n+1)/2; hexagonal = n(2n−1)). The ratio of perfect number to its Mersenne prime: N/(2^p − 1) = 2^(p−1) — a power of 2, not a golden ratio. However, the sum of reciprocals of divisors of a perfect number = 2 (harmonic mean of divisors is an integer), echoing the 2:1 octave ratio i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -