Estimation of Hailstone Radii using the Collatz Conjecture

There is an alleged connection between a much researched, yet unsolved math problem, the Collatz conjecture, and a quite common, damaging meteorological phenomenon: hailstorm. The relationship would be that, in sequences generated by Collatz algorithm, the way in which numbers rise and fall resembles hailstones going up and down inside a cloud, whence the name “Hailstone sequences”. The aim of this paper is two-fold: first, to use JavaScript to research on the Collatz conjecture with the perspective of a high school student. Our algorithm tested a generalized form of the conjecture for multiple primes (p =3, 5, 7) and signs (+, -). Our first Excel sheet (Collatz Java Code Analysis) tabulates the converging values, total stopping time and maximum value attained for different n, p, sign combinations. The Pearson correlation coefficient found between the initial value and, respectively, the total stopping time or the maximum value reached excluded any linear correlation. The second (and main) goal was to assess the hypothesis whether hailstones could indeed follow a Collatz-like function trajectory, studying the implication on the radii of them. Introducing the concept of conversion formula, we estimated the final radii for different functions (straight line, square-root, square, logarithmic, exponential), unit of measures (from Km to mm), and starting heights ranging from 4000m to 10000m, should the motion of hailstones behave like a Collatz function. Our second Excel file (Hailstone simulation new R old C) tabulates the radii achieved for different height, unit of measure and function combinations. In all but one case, we did not get radii believable in size, and reasonably randomly distributed. Instead, for the linear formula (in cm), the -test value between our estimated values and Nelson’s model values was 0.70. Hence, in this case we cannot dismiss the initial hypothesis and further steps should be taken.