# Difference between revisions of "Fermat prime"

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− | + | A '''Fermat prime''' is a [[prime number]] of the form <math>2^{2^n} + 1</math> for some [[nonnegative]] [[integer]] <math>n</math>. | |

− | + | A number of the form <math>2^{2^n} + 1</math> for nonnegative integer <math>n</math> is a [[Fermat number]]. The first five Fermat numbers (for <math>n=0,1,2,3,4</math>) are | |

<cmath> | <cmath> | ||

\begin{align*} | \begin{align*} | ||

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F_2 &= 2^{2^2}+1 = 17\\ | F_2 &= 2^{2^2}+1 = 17\\ | ||

F_3 &= 2^{2^3}+1 = 257\\ | F_3 &= 2^{2^3}+1 = 257\\ | ||

− | F_4 &= 2^{2^4}+1 = 65537 | + | F_4 &= 2^{2^4}+1 = 65537, |

\end{align*} | \end{align*} | ||

</cmath> | </cmath> | ||

− | Based on these results, one might conjecture (as did [[Fermat]]) that all Fermat numbers are prime. However, this fails for <math>n=5</math>: <math>F_5 = 2^{2^5}+1 = 4294967297 = 641 \cdot 6700417</math>. In fact the primes listed above are the only Fermat numbers known to be prime. | + | and each of these is a Fermat prime. Based on these results, one might [[conjecture]] (as did [[Fermat]]) that all Fermat numbers are prime. However, this fails for <math>n=5</math>: <math>F_5 = 2^{2^5}+1 = 4294967297 = 641 \cdot 6700417</math>. In fact, the primes listed above are the only Fermat numbers known to be prime. |

− | Fermat primes are also the only primes in the form <math>2^m+1</math>. | + | == Primes one more than a power of 2 == |

+ | Fermat primes are also the only primes in the form <math>2^m+1</math>. | ||

+ | |||

+ | ===Proof === | ||

+ | Suppose that <math>m</math> has an odd factor <math>a > 1</math>. For all odd <math>a</math>, we have by the [[Factor Theorem | Root-Factor Theorem]] that <math>x + 1</math> divides <math>x^a + 1</math>. Since this is true as a statement about [[polynomial]]s, it is true for every integer value of <math>x</math>. In particular, setting <math>x = 2^{m / a}</math> gives that <math>2^{m / a} + 1 | 2^m + 1</math>, and since <math>2^m + 1 > 2^{m / a} + 1 > 1</math>, this shows that <math>2^m + 1</math> is not prime. | ||

+ | |||

+ | It follows that if <math>2^m + 1</math> is prime, <math>m</math> must have no odd [[divisor | factors]] other than <math>1</math> and so must be a power of 2. |

## Latest revision as of 17:07, 25 August 2009

A **Fermat prime** is a prime number of the form for some nonnegative integer .

A number of the form for nonnegative integer is a Fermat number. The first five Fermat numbers (for ) are and each of these is a Fermat prime. Based on these results, one might conjecture (as did Fermat) that all Fermat numbers are prime. However, this fails for : . In fact, the primes listed above are the only Fermat numbers known to be prime.

## Primes one more than a power of 2

Fermat primes are also the only primes in the form .

### Proof

Suppose that has an odd factor . For all odd , we have by the Root-Factor Theorem that divides . Since this is true as a statement about polynomials, it is true for every integer value of . In particular, setting gives that , and since , this shows that is not prime.

It follows that if is prime, must have no odd factors other than and so must be a power of 2.