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Topic: prime powers (Read 997 times) |
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Christine
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prime powers
« on: Jul 18th, 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 18th, 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 21st, 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 21st, 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 21st, 2014, 10:22pm » |
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Now we just need to prove the Goldbach conjecture and we're done! 
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Wikipedia, Google, Mathworld, Integer sequence DB
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0.999...
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Re: prime powers
« Reply #5 on: Jul 22nd, 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.
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| « Last Edit: Jul 22nd, 2014, 2:44am by 0.999... » |
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Christine
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Re: prime powers
« Reply #6 on: Jul 23rd, 2014, 10:20am » |
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on Jul 21st, 2014, 4:33am, rmsgrey wrote:| At least one of p, q, r must be 2 for this to work at all. |
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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 24th, 2014, 5:17am » |
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on Jul 23rd, 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?
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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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