Are There Any Other Prime Solutions to the Equation x(x+1)+y(y+1)=z(z+1)?

[secretchord]

I want to solve equation x(x+1)+y(y+1)=z(z+1) 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?

 

[mfb]

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:

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))

 

[coolul007]

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.