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Are Prime Numbers Infinite?

Date: 06/18/98 at 23:22:24
From: Vincent
Subject: Prime numbers

How do we know the number of primes is infinite? 

Thank you!

Date: 06/19/98 at 07:56:33
From: Doctor Allan
Subject: Re: Prime numbers

Hi Vincent,

The proof depends on the fact that every natural number has a unique
factorisation into primes. I assume that you are aware of this fact. 
If not, please write again.

Now assume that there is only a finite number of primes and let's call 
these p_1, p_2, ... , p_m.

Look at the number (p_1)*(p_2)*...*(p_m) + 1

Because of the unique factorisation, this number can be factored into 
our primes p_1, ... , p_m or it is prime itself.

You should now convince yourself that neither of these options is 
possible. Therefore our assumption that there is only a finite number 
of primes is incorrect, proving that there is indeed an infinite 
number of primes.


-Doctor Allan,  The Math Forum   
Associated Topics:
High School Number Theory

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