Contents of this paper deals with details- some of them proved and rest of them have been discussed- may be in the form of conjectures* but mostly true without loss of generality. Centuries have lapsed but, we think, some of the natural phenomena have remained partially known and rigorous version, may be all time, remains a mystery that need mathematical revolution from the known version. We have, to our best, tried to analyze and attempted to harmonize the data pertaining to the hypotenuses of right triangles and explaining some basic features have tried to represent in statistical formats finally presented its analysis in graphical form. [* conjectures like – There cannot exist a natural number that possesses Five (5) distinct primitive triplets. [This will call conjecture 1]. Comment: We have, using computer program, searched to look for such a number up to 10,000 natural numbers but have not found any natural number possessing the property said above. Ceaseless efforts are still continued in the direction and we expect some flashes of success. In fact, there are many structural suggestions, not yet that leads towards a right successful track.