# Finding all prime solutions

by secretchord
Tags: prime, solutions
 P: 1 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?
 Mentor P: 11,601 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))