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. Sincerely, -Doctor Allan, The Math Forum http://mathforum.org/dr.math/ |
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