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Finding all prime solutions

  1. Feb 23, 2013 #1
    I want to solve equation [itex]x(x+1)+y(y+1)=z(z+1)[/itex] over primes. I found a solution x=y=2, z=3 and I have a hypothesis that this is the only solution over prime numbers, but I cannot prove it or find any other solution. Any hints, please?
     
  2. jcsd
  3. Feb 23, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    I can confirm that there are no other solutions for x,y below 100 000.
    The fact that 2 is in that one solution could be a hint that there are no other solutions, but I don't see a simple proof.

    Python:
    Code (Text):
    def isprime(n):
        for x in range(2, int(n**0.5)+1):
            if n % x == 0:
                return False
        return True

    primes= []
    for x in range(2,100000):
      if(isprime(x)):
        primes.append(x)

    for x in primes:
      if(x%1000==1):
        print("computing: ",x)
      for y in primes:
        zz=x*(x+1)+y*(y+1)
        z=1/2*(1+4*zz)**0.5-1/2
        if(round(z,0)==z and isprime(z)):
          print(x,y,round(z,0))
     
    Last edited: Feb 23, 2013
  4. Feb 23, 2013 #3
    The terms A(A+1) are twice the sum of a series, dividing by 2 we get an equation that says: I need to sums of series that add to a third sum. This is about as far as I got, May not be much help but is a different view.
     
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