# Finding all prime solutions

1. Feb 23, 2013

### 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?

2. Feb 23, 2013

### 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
3. Feb 23, 2013

### 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.