Factorizing Mersenne Numbers

sign

-

+

factor

850

2

-

1

850

2

-

1 = 3 × 11 × 31 × 251 × 601 × 1801 × 4051 × 43691 × 131071 × 91362251 × 5046718903451 × 9520972806333758431 × 26831423036065352611 × 4777345536534924905725989065906794483551790056167373849557595739795782900601 × 2069237502716464794985816105550982396339012259800336045348830659287429006970383760001800897298401

A number of the form

n

2

-1

is a Mersenne number, named after Marin Mersenne (1588–1648). If the resulting number is prime, then it is called a Mersenne prime, with

57885161

2

-1

(or M57885161) currently the largest known prime.

In 1925, Allan Cunningham published tables of factorizations for numbers of the form

n

a

-1

, including Mersenne numbers with exponents up to 600. The tables were later extended to exponents up to 1200 by Lehmer and Selfridge. On June 17, 2015, the last gap in the table was filled with the factorization of

991

2

-1

.

This Demonstration gives factorizations for all Mersenne numbers up to M1206. In binary, a Mersenne number is all ones, 1111...1111. After the factorization, the binary forms of the factors are concatenated to make a pattern.

Numbers of the form

n

2

+1

are also included, but a few cases like

983

2

+1

have not been completely factored. In these cases, the composite part is indicated in red.