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Version 12 |
| \textbf{Fermat Numbers.}\\ |
\textbf{Fermat Numbers.}\\ |
| The $n$-th Fermat number is defined as: |
The $n$-th Fermat number is defined as: |
| $$F_n=2^{2^n}+1.$$ |
$$F_n=2^{2^n}+1.$$ |
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| Fermat incorrectly conjectured that all these numbers were primes, |
Fermat incorrectly conjectured that all these numbers were primes, |
| although he had no proof. |
although he had no proof. |
| The first 5 Fermat numbers: $3, 5, 17,257,65537$ (corresponding to $n=0,1,2,3,4$) are all primes (so called Fermat primes) |
The first 5 Fermat numbers: $3, 5, 17,257,65537$ (corresponding to $n=0,1,2,3,4$) are all primes (so called Fermat primes) |
| Euler was the first to point out the falsity of Fermat's conjecture |
Euler was the first to point out the falsity of Fermat's conjecture |
| by proving that $641$ is a divisor of $F_5$. (In fact, $F_5=641\times6700417$). |
by proving that $641$ is a divisor of $F_5$. (In fact, $F_5=641\times6700417$). |
| Moreover, no other Fermat number is known to be prime for $n>4$, so now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many composite Fermat numbers or not. |
Moreover, no other Fermat number is known to be prime for $n>4$, so now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many composite Fermat numbers or not. |
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| One of the famous achievements of Gauss was to prove that the regular polygon of $m$ sides can be constructed with ruler and compass if and only if $m$ can be written as |
One of the famous achievements of Gauss was to prove that the regular polygon of $m$ sides can be constructed with ruler and compass if and only if $m$ can be written as |
| $$m=2^k F_{r_1}F_{r_2}\cdots F_{r_t}$$ |
$$m=2^k F_{r_1}F_{r_2}\cdots F_{r_t}$$ |
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where $k\ge 0$ and the other factors are distinct primes of the form $F_n$ (of course, $t$ may be $0$ here, i.e. $m=2^k$ is allowed).
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where $k\ge 0$ and the other factors are distinct primes of the form $F_n$.
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| There are many interesting properties involving Fermat numbers. For instance: |
There are many interesting properties involving Fermat numbers. For instance: |
| \[ |
\[ |
| F_m = F_0F_1\cdots F_{m-1}+2 |
F_m = F_0F_1\cdots F_{m-1}+2 |
| \] |
\] |
| for any $m\geq 1$, which implies that $F_m-2$ is divisible by all smaller Fermat numbers. |
for any $m\geq 1$, which implies that $F_m-2$ is divisible by all smaller Fermat numbers. |
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| The previous formula holds because |
The previous formula holds because |
| \[ |
\[ |
| F_m -2 = (2^{2^m}+1)-2 = 2^{2^m}-1 = (2^{2^{m-1}}-1)(2^{2^{m-1}}+1) = (2^{2^{m-1}}-1) F_{m-1} |
F_m -2 = (2^{2^m}+1)-2 = 2^{2^m}-1 = (2^{2^{m-1}}-1)(2^{2^{m-1}}+1) = (2^{2^{m-1}}-1) F_{m-1} |
| \] |
\] |
| and expanding recursively the left factor in the last expression gives the desired result. |
and expanding recursively the left factor in the last expression gives the desired result. |
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\bigskip |
| \textbf{References.}\\ |
\textbf{References.}\\ |
| Kr\'\i zek, Luca, Somer. \emph{17 Lectures on Fermat Numbers.} CMS Books in Mathematics. |
Kr\'\i zek, Luca, Somer. \emph{17 Lectures on Fermat Numbers.} CMS Books in Mathematics. |
