Fermat Primes are prime numbers of the form , where *n* is a non-negative integer and are named after the French Mathematician Pierre de Fermat who studied numbers of this form.

**If 2**^{n} + 1 is a prime, then *n* is a power of 2

^{n}+ 1 is a prime, then

*n*is a power of 2

For 2^{n} + 1 to be prime, then *n* must not contain an odd factor, or 2^{n} + 1 would be a factorable number of the form:

However, although this condition is necessary, it is not sufficient. Fermat conjectured that all the numbers of this form would be prime, however in 1732, Leonard Euler showed that F5 is a composite (4,294,967,297 = 641 * 6,700,417). Currently, we only know of 5 prime Fermat Numbers:

Pépin’s Test: Théophile Pépin showed that a Fermat number is prime if and only if:

This condition is both necessary and sufficient.

### Properties

**For n ≥ 1,**

This can be proved by induction:

*When n = 1: F0 + 2 = 3 + 2 = F1 *

*Assume: Fn = F0···Fn-1 + 2 is true*

*Then: F0···Fn + 2 *

* = F0···Fn-1· F**n + 2*

* = (Fn – 2)·Fn + 2*

* = (2 ^{2^n} – 1)(2^{2^n} + 1) + 2*

* = 2 ^{2^n+1} + 1 = Fn+1*

**For n ≥ 1,**

Proof: (Fn-1 -1)^{2} + 1 = (2^{2^n-1} +1 – 1)^{2 }+ 1 = *2 ^{2^n} *+ 1 = Fn

**No Fermat number is a perfect square.**

F0 and F1 (3 and 5 respectively) are clearly not perfect squares.

For Fn where n ≥ 2, Fn ≡ 7 (mod 10). However, only numbers that are congruent to 0, 1, 4, 5, 6, or 9 (mod 10) can be a perfect square.

**No two Fermat numbers share a common factor greater that 1.**

We will prove this by contradiction. Let’s suppose there exist Fi and Fj (where Fj > Fi) such that there exists an a > 1 that divides both of them. Another property of Fermat numbers that can be shown by induction is:

Hence, .

As Fi divided Fj, *a* also divides Fj-1 and thus F0···Fi···Fj-1. Then, *a* has to divide the difference Fj – F0···Fj-1, which equals 2 (as shown in property 1). It follows that a = 2, but all Fermat numbers are odd, therefore there is a contradiction and so no two Fermat numbers share a common factor greater that 1.

I have only covered a few properties satisfied by Fermat Primes, but this article is very comprehensive and lists a wide selection of their properties.

Little is known about Fermat numbers for large *n*, and in fact the following are all open problems:

- Is
*F*_{n}composite for all*n*> 4?*So far no primes have been found for n > 4.* - Are there infinitely many Fermat primes?
^{} - Are there infinitely many composite Fermat numbers?
- Does a Fermat number exist that is divisible by a perfect square other than 1
*i.e. not square-free*?

Hope you liked this more mathematics heavy post! Let me know what you think. M x

A very elegant proof, thank you for sharing

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Pretty nice blog…👌👌

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Thank you!

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You are very welcome… Please do make some time to visit my blog..

Thanks 😄

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I find Fermat primes interesting only since, towards the end of the book Disquisitiones Arithmeticae (“investigations of arithmetic”), Gauss proved that:

“A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes (including none).”

I you become “very” interested in Number Theory, try to read Disquisitiones Arithmeticae, following versions are available:

(1) Original Latin: http://edoc.hu-berlin.de/ebind/hdok2/h284_gauss_1801/pdf/h284_gauss_1801.pdf

(2) Spanish translation: http://cimm.ucr.ac.cr/da/

(3) English translation: http://yalebooks.co.uk/display.asp?k=9780300094732

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Thank you! I’ll definitely try to read it, although I think I’ll have to stick the the english one!

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