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In this paper, we investigate the possible scenarios in which a number does not satisfy the Collatz Conjecture. Specifically, we examine numbers which may have a looping Collatz reduction sequence as well as numbers which may lead to a diverging Collatz reduction sequence. In order to investigate these, we look at the parity of the numbers in a general Collatz reduction sequence. Further, we examine cases in which these parity cycles repeat themselves infinitely in the reduction sequence. Through the research conducted in the paper, we formulate a necessary condition for looping in the Collatz Conjecture. We also prove that if a number has a diverging reduction sequence, then it must generate an infinite non-repeating parity cycle. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the possible scenarios in which a number does not satisfy the Collatz Conjecture. Specifically, we examine numbers which may have a looping Collatz reduction sequence as well as numbers which may lead to a diverging Collatz reduction sequence. In order to investigate these, we look at the parity of the numbers in a general Collatz reduction sequence. Further, we examine cases in which these parity cycles repeat themselves infinitely in the reduction sequence. Through the research conducted in the paper, we formulate a necessary condition for looping in the Collatz Conjecture. We also prove that if a number has a diverging reduction sequence, then it must generate an infinite non-repeating parity cycle. [ABSTRACT FROM AUTHOR]

Reviewing the initial data coding to articles like Professor Beebe's is very much within the skillset of law journals. 40 NEW YORK UNIVERSITY JOURNAL OF INTELLECTUAL PROPERTY AND ENTERTAINMENT LAW VOLUME 10 FALL 2020 NUMBER 1 EDITORIAL, THE NEED FOR COLLECTIVE STANDARDS: VALIDATING RAW DATA IN LEGAL EMPIRICAL ANALYSIS ZACHARY J. BASS, ASHLEY C. ULRICH, RYAN B. MCLEOD, KEVIN QIAO, JOANNE DYNAK, LVXIAO CHEN, GARRETT HELLER, MINYOUNG RYOO, MAGDALENA CHRISTOFOROU, JESSE KIRKLAND, NEIL CHITRAO, PATRICK A. REED, AMANDA GONZALEZ BURTON, SIDDRA SHAH, NICHOLAS J. ISAACSON, JERRIT YANG * For some time, legal academia has experienced an increase in articles utilizing empirical analysis. Realistically, JIPEL and most other law journals do not have the resources to validate the data coding for 579 cases, especially where each case averages over 12 pages in length and is associated with over 100 data inputs. [Extracted from the article]