Fermat numbers

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Exploring Number Theory

Two previous posts (here and here) present an alternative proof that there are infinitely many prime numbers using the Fermat numbers. Specifically, the proof is accomplished by pointing out that the prime factors of the Fermat numbers form an infinite set. This post takes a look at Fermat numbers in more details.

A Fermat number is of the form $latex displaystyle F_n=2^{2^n}+1$ where $latex n=0,1,2,3,cdots$. The numbers grow very rapidly since each Fermat number is obtained by raising 2 to a power of 2. The first several terms are 3, 5, 17, 257, 65,537, 4,294,967,297. They are named after the French mathematicians Pierre de Fermat (1601-1665). He demonstrated that the first 5 Fermat numbers $latex F_0$, $latex F_1$, $latex F_2$, $latex F_3$, $latex F_4$ are prime and conjectured that all Fermat numbers are prime. In 1732, Euler provided a counterexample to the conjecture by showing that

    $latex displaystyle…

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