I asked the following valid question in a search engine:
“how to find the starting point of a Collatz sequence that passes through a given number”. The answer that it gave:
" To find the starting point of a Collatz sequence that passes through a given number, repeatedly apply the Collatz rule (divide by 2 if even, multiply by 3 and add 1 if odd) in reverse to the given number until you reach a number that cannot be further “un-Collatzed” - that number is the starting point of the sequence that includes the given number."
So we have a new verb for the forthcoming 2025 dictionaries, “un-Collatz” .
As a retired professor, I grieve for current professors who have to grade reports that are based on answers that students have squeezed out of AI engines.
In English, “any noun can be verbed.”
[In English, “any noun can be verbed.”
– Yes. AbsoBloomingCollatzly!
In reality, a restricted “un-Collatzing” is not only possible, but also useful. Given a current value y, we may attempt to find two integers x and j such that x is odd and 3x+1 = 2^j y . For example, with y= 49, we find 2^2(49)=196=3*65+1, so x=65 is a “parent” of y=49. Similarly, we can backtrack to 43 as a parent of 65 and
57 as a parent of 43. We cannot continue further back because 57 is a multiple of 3, and multiples of 3 cannot result from a calculation of the Collatz sequence since, for any integer n, 3n+1 can never be a multiple of 3. Remember, as well, that for odd n the expression 3n+1 will be even.
In general, any number can have multiple parents in a Collatz sequence. We need to specify additional criteria to select one of these as the parent.
For example, 9, 37, 149 and 597 are potential parents for 7. Which of the four would you choose, and why? Note also that each of these (except 9) is equal to 4 times the previous one plus 1.
Along the same lines, here are candidate parents for 11: 7, 29, 117, 469, 1877.