Infinite Number of PrimesDate: 04/14/98 at 15:44:14 From: Domenico Simonetti Subject: Numbers not divisible I'm looking for the greatest number not divisible calculated by some supercomputer or university's department or some mathematician, and want to know if they have or not discovered whether these numbers are or are not infinite. Can you help me please? Thank you very much! Date: 04/14/98 at 15:59:51 From: Doctor Kate Subject: Re: Numbers not divisible By "numbers not divisible," do you mean prime numbers? Numbers which are only divisible by 1 and by themselves (like 2, 3, 5, 7, 11, 13, 17...)? If so, then the largest one recently found is about 2 million digits long or something astronomical like that. You can read about The Largest Known Primes on the Web: http://www.utm.edu/research/primes/largest.html Primes *are* in fact infinite, and the proof was found a long time ago. It's a beautiful proof, so even if you *aren't* asking about primes, I'm going to tell you this gorgeous proof: Assume that primes are not infinite (i.e. there is a finite number of primes). We will come to a contradiction. If there is a finite number of primes, then we can multiply them all together. This will be a number divisible by every single prime. Now add 1. We now have a number that, when you divide it by any prime, will have a remainder of 1 (make sure you believe this). This number is a problem, because it can't be prime, and it isn't divisible by any prime. So what is it? All composites (divisible numbers) are divisible by some primes, so it can't be prime or composite. Therefore this number can't exist. This is a contradiction, so our original assumption (that prime numbers are finite) is wrong. Thus prime numbers are infinite in number. I suggest you look up primes in the Dr. Math FAQ for more information if you are interested. http://mathforum.org/dr.math/faq/faq.prime.num.html Hope this helps. -Doctor Kate, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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