# A New Way of Looking at the Collatz Conjecture

## Keywords:

Prime numbers, Congruence, Collatz Conjecture, Hailstone numbers## Abstract

The Collatz conjecture is named after a mathematician Lothar Collatz who introduced the conjecture in 1937. The Collatz conjecture which remains an unsolved problem in mathematics today, also known as the “3n + 1 conjecture”, explains about a sequence defined as follows: Start with any positive integer, the starting integer can be an even number or an odd number. Consider a number *n*, if the number *n* is even, divide it by 2. If the number *n* is odd, triple it and add one. Further, it states that regardless of the choice of *n*, after some iterations of the conjecture, the number no matter what value of a positive *n* is chosen, the sequence from the number chosen, projecting between lower and peak values will eventually attains the value of 1. Once reaching the value of 1 it will cycle through the values 1, 4, 2 indefinitely. The projections of numbers involved in this conjecture is sometimes referred to as “hailstone numbers” [1] because of the different projections of each number in multiple descents and ascents before reaching the number 1. I believe that the ideas here represent an interesting new approach towards understanding the Collatz conjecture’s sequences of numbers, especially the effect of the properties of even numbers divisible by 6, odd numbers divisible by 3, prime numbers and a set pattern in the projections in a certain arrangement. Following the theme, here in this article a defined pattern in the sequence of the projections which eluded mathematicians for years are discussed. This had also been computer tested and verified as true to 2^{68 }in 2020_{.}

## References

C.A. Pickover.” Hailstorm Numbers” Wonders of Numbers, Adventures in Mathematics, Mind, and Meaning. Oxford England: Oxford University Press, 2001, pp. 116 – 118.

J.C. Lagarias. The Ultimate Challenge: The 3x + 1 Problem. American Mathematical Society, 2010.

Terrance Tau. “The notorious Collatz conjecture-Terence Tau.” “Riemann Prize Week - Universita degli Studi delIlInsubria”. Internet: youtube, Oct. 30, 2021 [Jan. 29, 2022].

Richard K.Guy. “Permutation Sequences. Unsolved Problems in Number Theory.”Springer-Verlag, E17, p. 336–7. ISBN 0-387-20860-7. Zbl 1058.11001, 2004.

Lagarias, A Heuristic Argument, 1985.

Jared A. Grauer. “Analogy Between the Collatz Conjecture and Sliding Mode Control.” NASA Langley Research Center Hampton, Virginia .23681-2199. Jan. 01, 2021.

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*American Scientific Research Journal for Engineering, Technology, and Sciences*,

*88*(1), 91–98. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/7485

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