Math 2.0

Number theory Theorem

Fermat numbers are pairwise coprime

Builds on Fermat numberCoprime integers

Statement

Let mm and nn be distinct natural numbers. Then the Fermat numbers FmF_m and FnF_n are coprime.

Remarks

The proof uses F0F1⋯Fn−1=Fn−2F_0 F_1 \cdots F_{n-1} = F_n - 2. A common divisor of FmF_m and FnF_n with m<nm < n divides 22. Fermat numbers are odd, so the divisor is 11. The result is often called Goldbach's theorem, after Christian Goldbach. A consequence is a new proof that there are infinitely many primes: each FnF_n has a prime factor, and these primes are all distinct.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionFermat number, definition, no evidence levelFermat numberdefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionCoprime integers, definition, no evidence levelCoprime integersdefinitionFermat numbers are pairwise coprime, statement, no evidence levelFermat numbers are pa…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. Fermat number definition
  3. Greatest common divisor definition
  4. Coprime integers definition
  5. Fermat numbers are pairwise coprime statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Fermat_number, Basic properties
About this page
Tag05598J0Y8R
URIm2://05598J0Y8R
Aliasesgoldbach-theorem-fermat-numbers
Kindstatement, theorem
WritingW0 stub
MSC11A41
StatusThis copy of the site has no build data. Run m2 status 05598J0Y8R.
UsesFermat number, Coprime integers

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