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 2n + 1 is a prime, then n is a power of 2
For 2n + 1 to be prime, then n must not contain an odd factor, or 2n + 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.
- 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· Fn + 2
= (Fn – 2)·Fn + 2
= (22^n – 1)(22^n + 1) + 2
= 22^n+1 + 1 = Fn+1
- For n ≥ 1,
Proof: (Fn-1 -1)2 + 1 = (22^n-1 +1 – 1)2 + 1 = 22^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:
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 Fn 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?
Sources: 1 | 2
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