Fermat Number - Other Theorems About Fermat Numbers

Other Theorems About Fermat Numbers

Lemma: If n is a positive integer,

proof:

Theorem: If is an odd prime, then is a power of 2.

proof:

If is a positive integer but not a power of 2, then where, and is odd.

By the preceding lemma, for positive integer ,

where means "evenly divides". Substituting, and and using that is odd,

and thus

Because, it follows that is not prime. Therefore, by contraposition must be a power of 2.

Theorem: A Fermat prime cannot be a Wieferich prime.

Proof: We show if is a Fermat prime, then the congruence does not satisfy.

It is easy to show . Now write, . If the given congruence satisfies, then, therefore

Hence ,and therefore . This leads to

, which is impossible since .

A theorem of Édouard Lucas: Any prime divisor p of F_n = is of the form whenever n is greater than one.

Sketch of proof:

Let G_p denote the group of non-zero elements of the integers (mod p) under multiplication, which has order p-1. Notice that 2 (strictly speaking, its image (mod p)) has multiplicative order in G_p, so that, by Lagrange's theorem, p-1 is divisible by and p has the form for some integer k, as Euler knew. Édouard Lucas went further. Since n is greater than 1, the prime p above is congruent to 1 (mod 8). Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue (mod p), that is, there is integer a such that a2 -2 is divisible by p. Then the image of a has order in the group G_p and (using Lagrange's theorem again), p-1 is divisible by and p has the form for some integer s.

In fact, it can be seen directly that 2 is a quadratic residue (mod p), since (mod p). Since an odd power of 2 is a quadratic residue (mod p), so is 2 itself.