This method resulted in what was at the time "arguably the strongest result in the long history of the Conjecture." The sample must therefore be constructed keeping this in mind. Tao managed to create a method to select samples that would maintain their original character through each iteration, making it easier to keep track of how the sample would behave. The problem with such a method is that each number in the sample changes its character with each iteration - numbers change from even to odd and become multiples of different numbers. Similarly, Tao tried using a sample of positive integers to see whether the conjecture would hold. Tao's method was analogous to how you sample a population for an election: you have to pick a sample that is representative of the full population. The biggest recent breakthrough using traditional mathematical methods was in 2019, when mathematical supercelebrity Terence Tao of UCLA released a proof implying that about 99% of numbers greater than a quadrillion eventually reduce to values less than 200, which means 99% of numbers satisfy the Collatz Conjecture since all numbers less than 200 have manually been shown to end in the 1-4-2-1 loop. However, new computational methods may shed light on the problem. Posited in 1937, the Collatz Conjecture remains one of math's most well-known unsolved problems. The Collatz Conjecture states that regardless of which positive integer you start with, the sequence will end in the 1-4-2-1 loop. as you can see, it gets to 1 and then repeats the 4-2-1 sequence ad infinitum. Then repeat these steps with the result, forming a sequence of numbers. If it is even, divide it by 2 (giving n/2). If it is odd, multiply it by 3 and add 1 (giving 3n+1). Its statement is deceptively simple: take any positive integer, n. The Collatz Conjecture is one of maths' most notorious problems.
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