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+3) + G(G+3) = H(H+3)

The following UBASIC program was interrupted when the total of possible F and G was up to 239,285, and so the solutions found

(2,       3,       4)
(3,       7,       8)

are the only ones where F and G add up to less than 239,285.

list
   10   for Tot=1 to 1000000
   20          F=2
   30          while F<=Tot/2
   40           G=Tot-F
   50           if prmdiv(G)=G then
   60            :Lhs=F*(F+3)+G*(G+3)
   70            :Disc=9+4*Lhs
   80            :Sr=int(sqrt(Disc)+0.5)
   90            :if Sr*Sr=Disc then
  100             :H=(Sr-3)/2
  110             :if H=int(H) then print F,G,H
  115           F=nxtprm(F)
  120          wend
  130    next Tot
OK

run
 2       3       4.0
 3       7       8.0
Break in 80

?tot
 239285
OK

 

Posted by Charlie on 2008-04-19 12:55:35