Summary of The Simplest Math Problem No One Can Solve - Collatz Conjecture

00:00:00 - 00:20:00

The Collatz Conjecture is a problem in mathematics that is said to be incredibly difficult to solve. The problem involves determining whether or not a set of positive integers will eventually end up in a loop created by applying two rules. Professional mathematicians have been unable to solve the problem, but Jeffrey Lagarias is the world authority on the conjecture.

• 00:00:00 The Collatz conjecture is a problem in mathematics that is said to be incredibly difficult to solve. The problem involves determining whether or not a set of positive integers will eventually end up in a loop created by applying two rules. Professional mathematicians have been unable to solve the problem, but Jeffrey Lagarias is the world authority on the conjecture. Alex Kontorovich and Yakov Sinai looked at the paths the hailstone numbers take to get to one, and found that they are random.
• 00:05:00 The Simplest Math Problem No One Can Solve - Collatz Conjecture - is a sequence of numbers that, on average, shrinks rather than grows. This is due to the fact that every time you multiply an odd number by three and add one, it becomes an even number, and every time you do this, the next number in the sequence is 3/2 of the original value. There are two ways this could be false, and mathematicians are still trying to figure out which one it is.
• 00:10:00 The Collatz Conjecture states that there exists a number that starts a sequence that grows to infinity, but does not obey the same numerical gravity as all of the other numbers. Another possibility is that there exists a sequence of numbers that forms a closed loop, but all the numbers in this loop are unconnected to the main graph. However, as of 2019, one of the world's greatest living mathematicians, Terry Tao, has shown that 3x+1 obeys even stricter criteria, meaning that almost all numbers will end up smaller than any arbitrary function f of x so long as that function goes to infinity.
• 00:15:00 The Simplest Math Problem No One Can Solve - Collatz Conjecture is a problem in mathematics that has yet to be solved. Terry Tao's proof that almost all numbers obey a small number that is arbitrarily small is one piece of evidence supporting the conjecture, but proving that all numbers do is not the same thing. There are 10% of numbers between one and 100 that are perfect squares, 31 numbers between one and 1,000 that are perfect squares, and the fraction of numbers that are not perfect squares goes to one as X goes to infinity. But proving the conjecture is not easy, and there is a chance that it will never be proven true.
• 00:20:00 The Simplest Math Problem No One Can Solve is the Collatz Conjecture, which is a problem that is almost anyone can understand and play around with, but which is hard to figure out for ourselves because of the organic-looking structure of numbers. The video describes how Brilliant, a website and app that helps learners learn by engaging them in problem-solving, helps take learners' understanding to the next level.