Number of Primes is Infinite/Proof 5
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Theorem
The number of primes is infinite.
Proof
Aiming for a contradiction, suppose there exist $n$ prime numbers.
Consider the Fermat number $F_n$.
From Goldbach's Theorem, $F_n$ is coprime to each of $F_0$ to $F_{n - 1}$.
Therefore there must be a prime number which is a divisor of $F_n$ which is not a divisor of any of $F_0$ to $F_n$.
But, again from Goldbach's Theorem, each of $F_0$ to $F_{n - 1}$ is coprime to every other Fermat number.
So all the $n$ prime numbers must have been exhausted being used as prime factors of $F_0$ to $F_{n - 1}$.
So there must be more prime numbers than $n$.
The result follows by Proof by Contradiction.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $257$