Newsgroups: sci.math
Subject: Re: fun with primes
From: khill@vax1.umkc.edu
Date: 4 Apr 92 08:11:13 CST
 <1992Mar20.043143.22643@news.iastate.edu> <1992Mar20.050549.9731@cs.cornell.edu> <1992Mar29.202754.22814@umbc3.umbc.edu>
Organization: University of Missouri-Kansas City

In article <1992Mar29.202754.22814@umbc3.umbc.edu>, marklc@umbc4.umbc.edu (Mr. Mark L. Cohen; ACS (C)) writes:
> In article <1992Mar20.050549.9731@cs.cornell.edu> csuley@cs.cornell.edu (Christopher Suley) writes:
>>In article <1992Mar20.043143.22643@news.iastate.edu>
>>ddkilzer@iastate.edu (David D Kilzer) writes:
>>>Actually, there are a FINITE number of prime numbers.  I don't recall the last
>>>prime number, but I know it exists.
>>
>>Bzzt! There are an infinite number of primes.
>>Let's assume you have the largest prime P.
>>Compute P! + 1, and voila, a new prime.
> 
> 
> This is not true:
> 5 is prime...  5! = 120, so according to your trick, 121 is prime.  But
> of course 11^2 is 121, so it isn't.
> 
> Therefore, I don't think so.
> 
>    Mark Cohen    marklc@umbc4.umbc.edu   

	You are reading too much into the proof.  The proof never said that
121 was prime wrt all integers.  The proof just says that 121 is prime wrt
5, 3, and 2.  
	
	Follow the reasoning of the proof more carefully:
	(Let each variable be an element of the integers)

  ASSUMPTION:	*) There is a largest prime.  Call it 'p'!
  --------------------------------------------------------
  THEREFORE:	1) Let 'p' be the largest prime.
			-- this is a powerfull conclusion from the
			   assumption that states, for example, 11 
			   is not prime!!
            
  		2) Let q=(p)(p-1)(p-2)(p-3)...(3)(2)(1) + 1 = p!+1

		3) Now, q(mod p)==1, q(mod (p-1))==1, q(mod (p-2))==1.
		   In general q(mod z)==1, p<=z<=2.  More simply put,
		   q is not divisible by any integer z, p<=z<=2!
	
		4) Thus, q is relatively prime to all of the numbers we 
		   called primes in our assumption (*).  Therefore, q is a new
		   prime wrt to all previously allowed primes and our
		   assumption is false.  Hence 'p' is not the largest prime;
                   there is at least one more prime, 'q', greater than 'p'.

	This proof never said anything about q being relatively prime to all
primes as we know them, just to primes less than or equal to 5.  Remember,
by the original assumption 11 is not prime. 
	This proof works, not because it gives us the NEXT prime, but because 
it continually proves that our upper bound on the set of primes is greater
than we originally assumed.  So what if it skips a few primes in between, we
now know that there is at least one more prime w, 5<w<=121.  Remember, wrt to
all primes 'p', p<=5, 121 is prime!
 
-- 
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| Ken Hill              | The opinions in this letter are not mine.  They were |
| Amateur Mathematician | created by an evil computer with a mind of its own!  |
| Professional Paper boy|                                                      |
| KHILL@VAX1.UMKC.EDU   |   :)<-<   "HELP, I've fallen and I can't get up!"    |
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Newsgroups: sci.math