extension of

$abc$-conjecture to positive rational numbers

Let’s represent each natural number as the sequence of prime multiplicities

$\alpha (n,p)$. For example, the number

$36=2^2{3}^2$ is represented by the sequence

$2,2,0,0,0,\dots$. Similarly,

$15$ would be represented as

$0,1,1,0,0,0,\dots$.

We can describe the notion of relative primeness in terms of orthogonality of these sequences.

Let’s define a scalar product between two numbers as

$⟨a,b⟩=\sum _{p\in \text{primes}}\alpha (a,p)\alpha (b,p)$, where

$\alpha (n,p)$ is the multiplicity of prime

$p$ in number n. Using this definition, we can note that two natural numbers will have a scalar product of

$0$ exactly when they are coprime / relatively prime.

With this inner product thinking in mind, we can extend our consideration to positive rational numbers. Each positive rational number can be written as

$\frac{k}{l}$ where

$k$ and

$l$ are relatively prime. We can map this rational number to the sequence

$\alpha (r,p)-\alpha (q,p):p\in \text{primes}$. So, for example

$\frac{2}{3}$ would become

$1,-1,0,0,0,\dots$ and

$\frac{36}{25}$ would be represented as

$2,2,-2,0,0,\dots$.

We can similarly extend our notion of scalar product to rational numbers

$\frac{a}{b}$ and

$\frac{c}{d}$ as follows:

$$⟨\frac{a}{b},\frac{c}{d}⟩=\sum _{p\in \text{primes}}(\alpha (a,p)-\alpha (b,p))(\alpha (c,p)-\alpha (d,p))$$

Equipped with this product, we can generalise the notion of coprimality, and say that two rational numbers are ‘coprime’ if the scalar product, as defined above, is 0.

For example, the positive rational numbers

$\frac{2}{3}$ and

$6$ are “coprime” by this extended definition. I will write

$\frac{2}{3}\perp 6$ to denote this.

$pqr$ triples

With an extended notion of coprimality to positive rational numbers, we can try to extend the

$abc$ conjecture to positive rational numbers as follows:

We are looking for positive rational solutions to

$p+q=r$ where

$p\perp q$,

$q\perp r$ and

$q\perp r$ (where

$\perp$ means that the scalar product as defined above is zero).

Firstly, for natural numbers,

$a,b,c$ triples from the abc-conjecture are a special case of

$p,q,r$ triples. But are there any examples that aren’t with natural numbers?

The answer turns out to be positive. With a little bit of coding, it's possible to find a couple of simple examples:

$$\frac{1}{10}+\frac{2}{105}=\frac{5}{42}$$

$$\frac{5}{36}+\frac{3}{100}=\frac{38}{225}$$

In fact, these rational solutions are not that rare. Out of rational numbers with numerator and denominator bound by 300, 75% of the coprime solutions to

$a+b=c$ are rational, only 25% of them are integer.

generalising the conjecture

What would a generalisation of the abc-conjecture look like for these rational solutions. We have multiple ways of formulating it:

First, we have to quantify the size of the solution. We have a couple of options:

• We can use the height
  $H$ of a rational number
  $\frac{k}{l}$, defined as
  $max\{k,l\}$, where
  $k\perp l$. We may quantify the size of the rational solution
  $p,q,r$ as
  $\log (maxH(p),H(q),H(r))$.
• We may also use weighted absolute multiplicities. For a rational number
  $\frac{k}{l}$ we define this as the norm
  $${\Vert \frac{k}{l}\Vert }_{\text{WAM}}=\sum _p(\alpha (k,p)+\alpha (l,p))\ln p$$. This would make sense as this is a kind of weighted
  $L1$ norm-type quantity. This is clearly just the log of the product of denominator and numerator for the rational number.

Second, we also need something to replace the radical

$rad(abc)$ from the conjecture. Here, too, we have options:

• We can use the larger of the radical of the nominator, or denominator, using the height function as above, so the bound would use
  $rad(H(pqr))$
• Just like we defined
  $WAM$ as a sort of weighted L1 norm, we can define the log radical as a weighted L0 norm:
  
  $${\Vert \frac{k}{l}\Vert }_{\text{WNM}}=\sum _{p:\alpha (k,p)>0}\ln p$$

With these options in mind, there are multiple valid ways to extend the abc conjecture to positive rational solutions, which preserve the natural numbers as special cases. The one I like the most is:

Let

$p,q,r$ be a rational solution to

$p,q,r$ such that

$p\perp q,p\perp r,q\perp r$, then

$$max\{\Vert p{\Vert }_{\text{WAM}},\Vert q{\Vert }_{\text{WAM}},\Vert r{\Vert }_{\text{WAM}}\}>(1+ϵ)\Vert pqr{\Vert }_{\text{WNM}}$$

only finitely often for any

$ϵ>0$.

If we define quality along these lines, we will find that solutions with quality above 1 are indeed rare. The highest-quality triple whose maximal WAM is large is this one:

$$\frac{15}{26}+\frac{216}{65}=\frac{39}{10}$$

Here

$ln(216\cdot 65)=9.54$, thus 'b' is a large number, but these numbers are all made of the primes

$2,3,5,13$ whose logarithmic sum is only

$5.97$.

Questions:

1. Does this make any sense?
2. Does the orthogonality-based generalisation of coprimality have a name? I couldn't find out, annoyingly, but that might be because it's trivial or not interesting enough, or plain useless.
3. Which version of the extension makes most sense, if any of them do?