Math 2.0

Number theory Proposition

If 2^p − 1 is prime then p is prime

Builds on Mersenne numberPrime number

Statement

Let nn be a natural number. If 2n−12^n - 1 is a prime number, then nn is a prime number.

Remarks

The proof: if n=abn = ab with 1<a<n1 < a < n, then 2a−12^a - 1 divides 2n−12^n - 1, because x−1x - 1 divides xb−1x^b - 1 for x=2ax = 2^a. And 1<2a−1<2n−11 < 2^a - 1 < 2^n - 1. For example, 26−1=632^6 - 1 = 63 is divisible by 22−1=32^2 - 1 = 3 and by 23−1=72^3 - 1 = 7. The converse is false: 1111 is prime, but 211−1=23⋅892^{11} - 1 = 23 \cdot 89.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionMersenne number, definition, no evidence levelMersenne numberdefinitionPrime number, definition, no evidence levelPrime numberdefinitionIf 2^p − 1 is prime then p is prime, statement, no evidence levelIf 2^p − 1 is prime t…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Divisibility definition
  2. Mersenne number definition
  3. Prime number definition
  4. If 2^p − 1 is prime then p is prime statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Mersenne_prime, Theorems about Mersenne numbers
About this page
Tag07MMWAXMYD
URIm2://07MMWAXMYD
Kindstatement, proposition
WritingW0 stub
MSC11A41
StatusThis copy of the site has no build data. Run m2 status 07MMWAXMYD.
UsesMersenne number, Prime number

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