Determine all possible triplets (x,y,z) of prime numbers satisfying:

x² + 37*x*y = z³ + 1656

Prove that there are no others.

 10    for T=2 to 9999
 20      for Xord=1 to T-1
 30        X=prm(Xord)
 40        Yord=T-Xord
 50        Y=prm(Yord)
 60        Zcubed=X*X+37*X*Y-1656
 65        if Zcubed>0 then
 70         :Z=int(Zcubed^(1/3)+0.5)
 80         :if Z*Z*Z=Zcubed then
 90          :if prmdiv(Z)=Z then
100            :print X;Y;Z
110      next Xord
120    next

finds only

 x=29  y=2  z=11
 
Though I have no proof, the program uses primes up to the 9998th prime (=104717) for x and y, though only when their ordinal position in the primes adds up to 9999 or less. But all instances of x or y being up to the 5000th prime (48611) have been tested.

Posted by Charlie on 2014-03-12 15:14:27