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I was doing some research into the Collatz conjecture and the main problem as I see it is this: What does +1 do to a factorization?

Is there any patterns, theorems or recent research that tackles this question: How does adding one affect factorization?

What I though maybe easier (or much harder, who knows) since it is a more specific case directly relating to the Collatz: $3/2x+1/2$ how does that change the factorization of x for odd x.

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    For starters, none of the old factors will work. – 2017-01-08
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    That's very profound. – 2017-01-09
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    Lmao, no it isn't... – 2017-01-09
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    That must take some deep analysis into modern number theory, probably had to look at the zeta function of modular forms to get that result. – 2017-01-09
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    Im just kidding – 2017-01-09
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    $$\color{white}{\text{Lmao, if that's what the zeta function is for...}}$$ – 2017-01-09
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    Oh I think I know how you got it. Some topology. Studying Euler characteristics huh? xd – 2017-01-09
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    $\color{#e0e0e0}{\text{When my text starts disappearing, you should stop the jokes XD}}$ – 2017-01-09
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    Actually, I derived it using M-Theory, because that is $100\%$ relevant here :-P – 2017-01-09
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    Which version of M-theory? Can't you use this new age quantum physics to prove the earth is flat xddd – 2017-01-09
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    Man, I love joking with mathematicians and physicists, you guys are the only ones who understand anything. – 2017-01-09
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    So true...! :-( Need smarter friends IRL... – 2017-01-09

2 Answers 2

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Let's see,not only will the factors of one never be the same, but parity of different types of factors change 1 mod 3 can have any number of 1 mod 3 factors, and an even number of 2 mod 3 factors. A 2 mod three can have any number of 1 mod 3 factors, but requires an odd number of 2 mod 3 factors. For primes p, (3p+1)/2 can be prime. This actually interlinks the Goldbach and Collatz conjectures. Proof:

if $p+q=2n$ where $2n$ is of form $4x+2$, then $$(\frac{3}{2}p+1/2)+\frac{3}{2}q+1/2)=3n+1$$ which is possible $10=7+3\implies 16=11+5$ for example.

How this relates to factoring, is that in a Goldbach partition, neither can be a factor so it eliminates non-factors.

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This question is too broad.

The problem of understanding how the multiplicative properties of $n+k$ relate to that of $n$ is an extremely difficult problem, and you might argue that this is the reason behind the difficulty of some of the most difficult problems in number theory. Obvious examples include Collatz, Goldbach and the twin prime conjecture. Indeed, (in some sense) most progress on these problems come down to addressing this chief issue of understanding the multiplicative properties of $n$ and $n+k$, in a round-about way. So, looking into the extremely broad literature on this subject can help.

As far as I know nothing nontrivial is known that is elementary. The obvious $(n,n+k)=(n,k)$ is known, for example, and we can use this to see that $n,n+k$ can't share any common prime factors except those dividing $k$. Beyond this simple observation, much of the rest is research level maths.