The closely related function Col' which also divides 3n+1 by 2 in the odd case, is concisely represented by the 65-bit lambda calculus term λ1(λλλ31(λλ2(421)))(λλ1)1(λλ1) operating on Church numerals [1]. It starts from the pair of numbers n and 0 and then performs n iterations of swapping the numbers after incrementing the first. Its lambda diagram is
Does this have some significance for back propagation or something, or is it just an interesting trick of arithmetic? //not that it needs to have a technical use, it's still neat.
Collatz, busy-beavers, and algorithmic information theory are all related. To the extent they offer insight into the sparseness or density of irreducible complexity in the space of all computation.. this has many implications for what can be computed efficiently, what can be learned efficiently, program-synthesis, what can be analyzed "at a distance" without just trying it and potentially needing to wait forever, etc.
Whether it will say anything very significant practically or only philosophically is a different question. Maybe it is something like the discovery of transcendentals.. finding out that most of the number line won't have a tidy algebraic closed-form isn't exactly a make-or-break deal for the program of mathematics itself, and it also doesn't matter much to people who are doing engineering
Hailstone numbers has been a popular subject in computing circles since forever. Not much practical application, just a very simple, but curious construct.
The closely related function Col' which also divides 3n+1 by 2 in the odd case, is concisely represented by the 65-bit lambda calculus term λ1(λλλ31(λλ2(421)))(λλ1)1(λλ1) operating on Church numerals [1]. It starts from the pair of numbers n and 0 and then performs n iterations of swapping the numbers after incrementing the first. Its lambda diagram is
[1] https://github.com/tromp/AIT/blob/master/fast_growing_and_co...The paper (2019): https://arxiv.org/abs/1909.03562
Does this have some significance for back propagation or something, or is it just an interesting trick of arithmetic? //not that it needs to have a technical use, it's still neat.
Collatz, busy-beavers, and algorithmic information theory are all related. To the extent they offer insight into the sparseness or density of irreducible complexity in the space of all computation.. this has many implications for what can be computed efficiently, what can be learned efficiently, program-synthesis, what can be analyzed "at a distance" without just trying it and potentially needing to wait forever, etc.
Whether it will say anything very significant practically or only philosophically is a different question. Maybe it is something like the discovery of transcendentals.. finding out that most of the number line won't have a tidy algebraic closed-form isn't exactly a make-or-break deal for the program of mathematics itself, and it also doesn't matter much to people who are doing engineering
Hailstone numbers has been a popular subject in computing circles since forever. Not much practical application, just a very simple, but curious construct.
youtube link https://youtu.be/k-dtx8s2ehM