Determine all possible positive integer triplet(s) (F, G, H) such that each of F and G is a prime with F ≤ G, and:
F(F+2) + G(G+2) = H(H+5)
Looking at the lists that Charlie and Larry generated, some solutions popped out at me:
3 19 18
7 67 66
13 199 198
37 1447 1446
61 3847 3846
73 5479 5478
139 19603 19602
157 24967 24966
I noticed all these have H=G-1, then a little poking I made a generator: {F,G,H} = {F, (F+1)^2+3, (F+1)^2+2} This satisfies the problem for any F (ignoring primality requirements)
The OEIS has Sequence A350676: Primes p such that p^2 + 2*p + 4 is prime. Any member of A350676 can be substituted for F to create an entire class of solutions to the problem.
A set of solutions
