The set of primes as an algebraic structure Let denote the set of all prime numbers, and let be an algebraic structure over (so for example, let be the group where is a (currently unknown) binary operation). We can do a lot with this algebraic structure, and the more structure it has, the more theorems we can endow on the prime numbers. However, finding such an algebra is not easy. For it to be useful, it must have a lot of structure, and a binary operation that is closed over the primes is rarely associative, let alone commutative, distributive, or idempotent. So what do you guys think? Is such a binary operation within our grasp? Would an algebraic structure of this type be applicable in any way? Please share your thoughts on the subject. 
Re: The set of primes as an algebraic structure Sure, here's an example. Let S(p) denote the prime succeeding p and P(p) denote the prime preceding p, if such a number exists. So P(5) = 3, P(11) = 7, and so forth. Then let and So for example 3*7 = S(2*7) = S(7) = 11. 
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Re: The set of primes as an algebraic structure The rules you stated were 1. 2*p=p*2=p 2. p*S(q)=S(q)*p=S(p*q) From these rules, 3*7=S(3*5)=S(15)=13 
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As it happens the operation is commutative, so it's faster to decrease the smaller number: 3*7 = 7*3 = S(7*2) = S(7) = 11. 
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I'm going to have to think about this for a while. Anyone else, please post your thoughts. :) 
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To compute p*q, first replace 11 with 13 if p or q is 11, and vice versa (replace 13 with 11 if p or q is 13). Then compute the operator described above. Finally, swap 11s and 13s as in the first step. Clearly this has the same structure as the first one, but it orders the primes 2, 3, 5, 7, [color=#800000]13, 11[/color], 17, 19, ... giving essentially the same result but with different names. I have a more complicated operation in mind but I'm not sure how to describe it. Here are some values: 13*3 = 11 7*7 = 2 17*23 = 47 3*13 = 11 2*5 = 5 5*2 = 5 19*3 = 17 3*7 = 5 13*5 = 19 7*29 = 31 2*13 = 13 11*17 = 5 2*47 = 47 5*7 = 3 59*43 = 113 
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