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I have two related questions that I can't seem to find any literature on:

1) What can be said about $\tau(a+b)$ knowing $\tau(a)$ and $\tau(b)$ (where $\tau(n)$ is the number of positive divisors of $n$)?

2) I have a feeling the answer to the previous is "nothing" because of the same types of issues surrounding the ABC-conjecture. If that's the case, are there any statements about $\tau(a+b)$ that are equivalent to, imply, or are implied by the ABC-conjecture?

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Well the existence of infinitely many solutions of $(\tau(a),\tau(b),\tau(a+b)) = (2,2,2)$ is equivalent to another famous open problem... –  Noam D. Elkies Aug 9 '12 at 2:56

1 Answer 1

up vote 5 down vote accepted

I agree that the answer to the first question should be more or less nothing. However, I disagree that this is for 'the same types of issues surrounding the ABC-conjecture', and so I highly doubt there is much of a direct link.

The ABC-conjecture is very different in that (in a certain sense) it says something on the actual divisors of the involved numbers, not merely on their number.

The reason why in my opinion the answer to the first question is essentially nothing is that simply there should be little meaninful to say, at least on a quantitative level (relating the respective approx. sizes).

To wit take $b=1$. If $a$ is a Sophie Germain prime then $a+b$ will only have $4$ divisors, but it is not at all true that $a+1$ has few prime divisors for any $a$ prime (ie any $a$ with $\tau(a)=2$).

Or, it is kown that in some sense there is 'nothing special' regarding divisors counts (except for evenness issues) of $p-1$ for $p$ prime relative to $n-1$ for $n$ any number.

(See also this MO question Factors of p-1 when p is prime. ; and I believ, but only checked briefly, the paper mentioned in a comment also goes roughly speaking in this direction; the result on something relating $\tau(n)$ and $\tau(n+a)$ coinciding with the prediction one has if considering the two 'independently').

And $b=1$ is not really special in that regard.

Or it is known that many (and if Goldbach conj. is true, all except one) even number can be written as sum of two primes, so (essentially) whatever even $n$ you consider you can write it as $n=a+b$ with $\tau(a)=\tau(b)=2$, so $\tau(a)=\tau(b)=2$ implies really (next to) nothing on $\tau(a+b)$.

So, in brief, yes there should be very little to be said (as regards to connecting these values), yet no this is not much related to ABC, in particular it is not that things are not known because they are diffcult but there is actually hardly any connection between $\tau(a+b)$ and $\tau(a)$, $\tau(b)$ on a quantitative level.

(If one were to investigate specific restrictions on the $\tau$, in particular such that they force/are equivalent too the number being a high power then the type of questions become a bit different and then there is I think some link to ABC; but then to me they are not really questions on $\tau$ but 'something else' encoded via $\tau$.)

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