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Can 4 distinct prime numbers be related in such a way?

  1. Jul 12, 2013 #1
    Hi everyone,

    I've been bumping on this problem for a while and wondered if any of you had any clue on how to approach it. My question is whether the following equality is possible for 4 distinct prime numbers :

    PxPy + Pw = PwPz + Px

    where Px, Py, Pw, Pz are odd prime numbers, and each of them is unique, thus not equal to the others.

    If impossible, how would you prove that ?


  2. jcsd
  3. Jul 12, 2013 #2
    An equivalent way of phrasing the question:

    Do there exist k,p,q with p,q distinct primes such that kp+1, kq+1 are also primes?

    [My p,q correspond to your Px,Pw.]
  4. Jul 12, 2013 #3
    Oh, it's not exactly equivalent. If p<q, then I need to further require that kp+1 is distinct from q.
  5. Jul 12, 2013 #4


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    How is that equivalent?
    With k=2, there are many such primes p,q. Examples: p=3, q=5.
    I don't see how we can construct the equality from the first post based on them.

    This is possible with other even k, too. Your k cannot be odd, otherwise kp+1 and kq+1 would be even and not prime.

    PxPy + Pw = PwPz + Px can be written as Px(Py-1) = Pw(Pz-1)
    As Px and Pw are distinct primes, (Pz-1) has to be divisible by Px and (Py-1) has to be divisible by Pw. As both (Pz-1) and (Py-1) are even, Pz>2Px and Py>2Pw.
  6. Jul 12, 2013 #5

    I also would add the following conditions:

    Since the equality Px(Py-1) = Pw(Pz-1) needs to reflect 1 integer that has a unique prime factorization, (Py-1) is an even number with prime factor Pw and (Pz-1) is an even number with prime factor Px.

    Still struggle to get to either a proof or a counterexample.
  7. Jul 12, 2013 #6


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    You seem to be almost there.

    [tex]\frac{x}{w} = \frac{z-1}{y-1}[/tex]

    Look at the right hand side and think about the prime decompositions of z-1 and y-1.
  8. Jul 12, 2013 #7
    Thank you very much for helping me like that I really appreciate.

    Not being a mathematician, here's what I could come up with, tell me if that makes sense.

    From your equation and from knowing that z-1 is even and has prime factor x, and that y-1 is even and has prime factor w, I get:

    [tex]\frac{x}{w} = \frac{2xA}{2wB}[/tex]

    where A and B are whatever remaining factors left except 2, x and w in the prime decompositions of the right side of the equation.

    Removing the 2 and multiplying each side by w and dividing each side by x, we get

    [tex]\frac{1}{1} = \frac{A}{B}[/tex]

    Therefore A = B, that is the only factor that is different in the prime decomposition of z-1 and y-1 is w and x.

    I know I must be very close to it, but I still can't see how exactly I will prove something on the original question. This tells me a lot about z-1, but what about z and y ?
  9. Jul 12, 2013 #8


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    Just test some prime numbers with Px(Py-1) = Pw(Pz-1) and the condition I found, there is an example with very small numbers.
  10. Jul 12, 2013 #9
    These two lines together should do something...
  11. Jul 12, 2013 #10


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    Please show how you can convert examples for your formulas (like the one I gave) to examples for the original problem.

    Those vague suggestions are not useful.
  12. Jul 12, 2013 #11
    I tried a couple and can't find the example. You found some without using 2, which isn't odd ?

    3*(13 - 1),5*(11-1)
    5*(17 - 1),7*(11-1)
    5*(17 - 1),7*(13-1)
    3*(17 - 1),7*(13-1)
    3*(17 - 1),5*(13-1)
    3*(23 - 1),5*(13-1)
    3*(29 - 1),5*(19-1)

    and many other all gave me inequalities, although I'd love to know what the counterexample is.



    Ahh my script was somewhat misprogramed.

    Here is the list of such sets of primes with primes below 29

    3 11 5 7
    3 23 11 7
    3 29 7 13
    5 7 3 11
    7 13 3 29
    11 7 3 23

    This opens up some more mysteries for me as I now have to find some characteristics of those sets of primes, thanks to all of you!
    Last edited: Jul 12, 2013
  13. Jul 13, 2013 #12


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    Try to prove that 3,5,7,11 is the smallest example. You are not yet understanding this relationship.

    I think enough help has been given here, let's leave JFGariepy to sort this out.
  14. Jul 13, 2013 #13
    Hum well the fact that 3 5 are the smallest primes to start with and that the other primes have to be at least double of 3 and 5 guarantees that this is the smallest pair.

    As for the understanding, I think I do understand that this is the set of primes where the difference between the 1st and 3rd primes is 2 times smaller (or a power of 2 smaller) than the difference between the 2nd and 4th primes. Humm that only holds for small numbers. Will have to work on that.
    Last edited: Jul 13, 2013
  15. Jul 13, 2013 #14


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    3,5,7,11 (=the example I meant) are the four smallest odd primes. How could there be a smaller example?

    He did that already :confused:.
  16. Jul 13, 2013 #15


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    I see the word "odd" now. I missed it before because the title does not use it and it even says "4 distinct prime numbers" in the quote above. If we are talking about odd primes then obviously 3,5,7,11 is the smallest case.
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