The Collatz conjecture involves a sequence where a number is divided in half if even or multiplied by three and added one if odd. It is assumed that all positive integers eventually end up in a loop that ends in 1, but this remains unproven.
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If you complete the whole numbers at the prime two the collatz iteration extends to this enlargement.
Then the even step “taking out the power of two” increases the natural “two“size and the second half step “multiplying by three“
preserves the natural “two” size
while the remaining second half step “adding one” decreases the natural “two” size.
Lunch comment by Alain Connes at IHES almost fifty years ago which continued to a proof of the Collatz conjecture for almost all two adic integers relative to the natural measure.
Thus
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The Collatz Conjecture is unsolved, not unsolvable.
It’s not “unprovable”, just “unproved”. And it’s not “assumed”, it’s “conjectured”. Mathematicians are picky about terminology.
Is this a joke? I am no mathematician, and so I can’t speak to the arguments on terminology, but I do know that (1) any even integer with an absolute value greater than two may be divided by two to produce another even integer with a lesser absolute value, (2) any two odd integers multiplied together will produce another odd integer, (3) all odd integers have two even integers adjacent to them (e.g. if |x%2|==1 then (x+1)%2==0), (4) the product or quotient of any two numbers can be determined to be positive or negative based on the count of signs such that provided that the number of positive terms is an even number, the resulting sign will always be positive, otherwise the resulting sign will be negative. Therefore, the conjecture only works for Natural numbers (positive integers excluding zero), and does work for all of them in a way so predictable that an algorithm could be written to shortcut the loop and predict instead the number of steps required to reach 1 from any given Natural number. Wherefore is this conjecture considered unproven? All the axioms are there, well-known, and quite self-evident, and all the more so with the epoch of calculus.