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Algebra and Primes (Posted on 2013-12-25) Difficulty: 2 of 5
Find all triples (x, y, z) of positive integers such that x + y + z - 3xyz = p

where p is a prime greater than 3

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.0000 (3 votes)

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Charlie's proof | Comment 2 of 3 |
let f(x,y,z) = x + y + z - 3xyz where x,y and z are positive integers

Then f(a,a,a+1) = 3a+1
        f(a,a,a-1)  = 3a-1

The minimum value of f is f(1,1,2) = 4

So, we can easily make the function equal any integer greater than 3 that is not divisible by 3.  No primes greater than 3 are divisible by 3, so we can make the function equal any prime.

There are probably many other ways to calculate many primes, and probably even ways to make multiples of 3.

Edited on December 25, 2013, 2:27 pm
  Posted by Steve Herman on 2013-12-25 14:19:40

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