In my research on right triangles I have found an interesting observation. It would be better if I explained it to you with an example. Please keep in mind that I am talking about integer right-triangles. Consider the triple 3,4,5. The sum of the (successor of 3) and 4, is 4 + 4 = 8. Here, 8 is a number which is the leg of more than 2 triangles: 6,8,10 and 8,15,17 etc. Again add 8 to 8. 8+8=16. 16 is the leg of another two right triangles (or probably many): 12,16,20 and 16,30,34. Again add 8 to 16. 8+16=24. 24 is the leg of two (probably many) triangles: 7,24,25 and 18, 24, 30.
This keeps going on and on for any triple you take and follow this procedure:
Add the successor of the smallest leg to the greater leg. Take it as x.
...(((X+x)+x)+x).... Are numbers which are definitely the legs of different right triangles. Is there any generalised proof for this or is there one triple which doesn't satisfy the procedure? My research tells me that there aren't any. I thank you for your help in advance.