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Topic: prime powers (Read 990 times) 

Christine
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prime powers
« on: Jul 18^{th}, 2014, 2:15pm » 
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I found many examples of the sum of two primes equal to a prime power p + q = r^n ............(p, q, r) are primes Is the number of solutions finite or infinite?


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Grimbal
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Re: prime powers
« Reply #1 on: Jul 18^{th}, 2014, 2:49pm » 
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If q is 2, and n 1, it is the prime pair conjecture.


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rmsgrey
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Re: prime powers
« Reply #2 on: Jul 21^{st}, 2014, 4:33am » 
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At least one of p, q, r must be 2 for this to work at all.


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dudiobugtron
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Re: prime powers
« Reply #3 on: Jul 21^{st}, 2014, 5:43pm » 
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When r is 2, then the existence of solutions for every n follows from Goldbach's Conjecture.


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towr
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Re: prime powers
« Reply #4 on: Jul 21^{st}, 2014, 10:22pm » 
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Now we just need to prove the Goldbach conjecture and we're done!


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0.999...
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Re: prime powers
« Reply #5 on: Jul 22^{nd}, 2014, 2:43am » 
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There's a chance Zhang's recent result on the distribution of the primes numbers could help, though the result and this riddle aren't exactly the same.

« Last Edit: Jul 22^{nd}, 2014, 2:44am by 0.999... » 
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Christine
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Re: prime powers
« Reply #6 on: Jul 23^{rd}, 2014, 10:20am » 
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on Jul 21^{st}, 2014, 4:33am, rmsgrey wrote:At least one of p, q, r must be 2 for this to work at all. 
 p + q = r^n (p, q, r) are primes if p > 2, q > 2, (p,q) are odd primes, then r = 2 could you please explain why?


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rmsgrey
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Re: prime powers
« Reply #8 on: Jul 24^{th}, 2014, 5:17am » 
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on Jul 23^{rd}, 2014, 10:20am, Christine wrote: p + q = r^n (p, q, r) are primes if p > 2, q > 2, (p,q) are odd primes, then r = 2 could you please explain why? 
 towr's already hit the key point, but: If p, q, and r are all odd, then r^n is also odd, but the sum of two odd numbers is an even number. Contradiction. Therefore at least one of p,q,r must be even (in fact, either one or all three of them must be) but they're also prime, so at least one must be an even prime, and, since all even numbers are divisible by two, the only one that's prime is two itself. There's a solution where p, q, r, n are all 2, and solutions where exactly one of p, q, r is 2, and no other solutions.


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