Fermat Number - Basic Properties

Basic Properties

The Fermat numbers satisfy the following recurrence relations:

for n ≥ 1,

for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common factor. To see this, suppose that 0 ≤ i < j and F_i and F_j have a common factor a > 1. Then a divides both

and F_j; hence a divides their difference, 2. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each F_n, choose a prime factor p_n; then the sequence {p_n} is an infinite sequence of distinct primes.

Further properties:

• The number of digits D(n,b) of F_n expressed in the base b is

(See floor function).

• No Fermat number can be expressed as the sum of two primes, with the exception of F₁ = 2 + 3.
• No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime.
• With the exception of F₀ and F₁, the last digit of a Fermat number is 7.
• The sum of the reciprocals of all the Fermat numbers (sequence A051158 in OEIS) is irrational. (Solomon W. Golomb, 1963)