Goldbach's Theorem
Theorem
Let $F_m$ and $F_n$ be Fermat numbers such that $m \ne n$.
Then $F_m$ and $F_n$ are coprime.
Proof 1
Aiming for a contradiction, suppose $F_m$ and $F_n$ have a common divisor $p$ which is prime.
As both $F_n$ and $F_m$ are odd, it follows that $p$ must itself be odd.
Without loss of generality, suppose that $m > n$.
Then $m = n + k$ for some $k \in \Z_{>0}$.
| \(\ds F_m - 1\) | \(\equiv\) | \(\ds -1\) | \(\ds \pmod p\) | as $p \divides F_m$ | ||||||||||
| \(\ds F_n - 1\) | \(\equiv\) | \(\ds -1\) | \(\ds \pmod p\) | as $p \divides F_n$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {F_n - 1}^{2^k}\) | \(\equiv\) | \(\ds -1\) | \(\ds \pmod p\) | Fermat Number whose Index is Sum of Integers | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {-1}^{2^k}\) | \(\equiv\) | \(\ds -1\) | \(\ds \pmod p\) | Congruence of Product | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1\) | \(\equiv\) | \(\ds -1\) | \(\ds \pmod p\) | Congruence of Powers | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0\) | \(\equiv\) | \(\ds 2\) | \(\ds \pmod p\) |
Hence $p = 2$.
However, it has already been established that $p$ is odd.
From this contradiction it is deduced that there is no such $p$.
Hence the result.
$\blacksquare$
Proof 2
Let $F_m$ and $F_n$ be Fermat numbers such that $m < n$.
Let $d = \gcd \set {F_m, F_n}$.
From the corollary to Product of Sequence of Fermat Numbers plus 2:
- $F_m \divides F_n - 2$
But then:
| \(\ds d\) | \(\divides\) | \(\ds F_n\) | Definition of Common Divisor of Integers | |||||||||||
| \(\, \ds \land \, \) | \(\ds d\) | \(\divides\) | \(\ds F_m\) | (where $\divides$ denotes divisibility) | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds d\) | \(\divides\) | \(\ds F_n - 2\) | as $F_m \divides F_n - 2$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds d\) | \(\divides\) | \(\ds F_n - \paren {F_n - 2}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds d\) | \(\divides\) | \(\ds 2\) |
But all Fermat numbers are odd, so:
- $d \ne 2$
It follows that $d = 1$.
The result follows by definition of coprime.
$\blacksquare$
Source of Name
This entry was named for Christian Goldbach.
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Further Exercises $9$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $257$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Fermat number (P. de Fermat, 1640)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Fermat number (P. de Fermat, 1640)
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- 2001: Michal Křížek, Florian Luca and Lawrence Somer: 17 Lectures on Fermat Numbers: Theorem $4.1$