September 1st, 2017, 11:56 AM  #51  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Quote:
 
September 1st, 2017, 12:13 PM  #52  
Senior Member Joined: May 2016 From: USA Posts: 881 Thanks: 353  Quote:
My position has been quite consistent although my very first post was poorly worded. If we assume that 5 is not prime, we are dealing with a different arithmetic (or a different definition of prime). Without further specification of what is intended, we do not know whether your original statement that ALL primes would be of the form 6n + 1 if 5 were not prime can be validated. It obviously cannot be validated for any arithmetic in which 5 is not prime because you yourself gave a counterexample. I agree with mashcke on two points. First, from a false premise, anything can be deduced. Second, an argument that both denies and relies on the rules of normal arithmetic is not interesting. Where I deviated from him was that I actually was interested in seeing where the supposition of an arithmetic where 5 was composite might go. Your mod 11 arithmetic seems to fit the bill, but, alas, not all primes in that arithmetic are of the form 6n + 1.  
September 1st, 2017, 12:17 PM  #53 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 112 Thanks: 35 Math Focus: Algebraic Number Theory, Arithmetic Geometry  
September 1st, 2017, 12:40 PM  #54 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  
September 1st, 2017, 01:13 PM  #55 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 112 Thanks: 35 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Just as we exclude $1$ from being a rational prime and $1,1,i,i$ from being primes in the ring of Gaussian integers, we exclude units from being primes in integral domains in general. Ultimately, it is just convention, but it does have advantages (and no significant drawbacks). This way, a lot of results are much easier to state, such as unique factorisation (which only holds in certain rings!) and various things relating prime ideals and prime elements. Note that fields are just nontrivial commutative rings in which all nonzero elements are units
Last edited by cjem; September 1st, 2017 at 01:50 PM. 

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