Math 2.0

statement theorem

Fermat numbers are pairwise coprime

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.

Explanations

L1 human

No two Fermat numbers share a prime factor. For example, F2=17F_2 = 17 and F3=257F_3 = 257 are coprime, and F5=641⋅6700417F_5 = 641 \cdot 6700417 has no factor in common with 33, 55, 1717, 257257 or 6553765537.

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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
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  1. Divisibility definition
  2. Fermat number definition
  3. Greatest common divisor definition
  4. Coprime integers definition
  5. Fermat numbers are pairwise coprime statement

Sources

  1. Web https://en.wikipedia.org/wiki/Fermat_number, Basic properties

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