Editing Fermat number (section)

==Basic properties==
The Fermat numbers satisfy the following [[recurrence relation]]s:

:<math>
F_{n} = (F_{n-1}-1)^{2}+1</math>

:<math>
F_{n} = F_{0} \cdots F_{n-1} + 2</math>

for ''n'' ≥ 1,

:<math>
F_{n} = F_{n-1} + 2^{2^{n-1}}F_{0} \cdots F_{n-2}</math>

:<math>
F_{n} = F_{n-1}^2 - 2(F_{n-2}-1)^2</math>

for {{nowrap|''n'' ≥ 2}}. Each of these relations can be proved by [[mathematical induction]]. From the second equation, we can deduce '''Goldbach's theorem''' (named after [[Christian Goldbach]]): no two Fermat numbers [[coprime|share a common integer factor greater than 1]]. To see this, suppose that {{nowrap|0 ≤ ''i'' < ''j''}} and ''F''<sub>''i''</sub> and ''F''<sub>''j''</sub> have a common factor {{nowrap|''a'' > 1}}. Then ''a'' divides both

:<math>F_{0} \cdots F_{j-1}</math>

and ''F''<sub>''j''</sub>; hence ''a'' divides their difference, 2. Since {{nowrap|''a'' > 1}}, this forces {{nowrap|1=''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''<sub>''n''</sub>, choose a prime factor ''p''<sub>''n''</sub>; then the sequence {{mset|''p''<sub>''n''</sub>}} is an infinite sequence of distinct primes.

===Further properties===
* No Fermat prime can be expressed as the difference of two ''p''th powers, where ''p'' is an odd prime.
* With the exception of ''F''<sub>0</sub> and ''F''<sub>1</sub>, the last decimal digit of a Fermat number is 7.
* The [[sums of reciprocals|sum of the reciprocals]] of all the Fermat numbers {{OEIS|id=A051158}} is [[irrational number|irrational]]. ([[Solomon W. Golomb]], 1963)