How do we know there are an infinite number of prime numbers? How do we know that once you get to a certain point, they don’t all end up as composite numbers?
In: Mathematics
Simple mathematical proof by contradiction.
Assume that there are a finite number of prime numbers. Then by definition, there is a largest prime, and every number larger than the largest prime is divisible by at least two of the primes.
Construct the number (N say) formed by multiplying all the primes together, then adding one. N is bigger than all the primes, but is not divisible by any of them. Contradiction.
Therefore our assumption that there are a finite number of primes is wrong.
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