The abc conjecture

The "abc conjecture" is a conjecture in number theory, proposed by David Masser and Joseph Oesterlé in 1985. It started to receive publicity in 2012, when Shinichi Mochizuki claimed to have proved it, in a 512-page paper. If the abc conjecture is true, it has consequences for many other results, including Fermat's Last Theorem.

Statements of conjecture

So what does the abc conjecture say? According to Wikipedia:

Given three positive integers, a, b and c that are relatively prime and satisfy a + b = c: if d denotes the product of the distinct prime factors of abc, ... d is usually not much smaller than c.
Wikipedia then gives a formal definition:
For every ε > 0, there exist only finitely many triples (a, b, c) of positive pairwise-coprime integers, with a + b = c, such that c>rad(a*b*c)^(1+ε)
where rad(n) is the product of the distinct prime factors of n.

According to Nature (October 7, 2015) it says

if a lot of small primes divide a and b then only a few, large ones divide c.

Some notation

In this page, we will denote rad(a*b*c) by d. We will call a triple (a,b,c) such that d<c a "hit".

Is the conjecture surprising?

The claim, at least in the popularised versions of the conjecture quoted above, is that d is usually greater than c, and rarely much smaller. This seems unremarkable. There aren't many small primes. And a,b,c are all pairwise-coprime; no more than one of them can be a multiple of a power of 2, or of 3, etc. We ought to expect hits to be quite rare.

And are they rare? It's quite easy to think of triples with d<c:

To me, it feels as if the numbers are conspiring to give more hits than should be expected.

A controlled experiment

So I wanted to test the conjecture:

For a pairwise-coprime triple (a,b,c) with a=b+c, d<c less often than we might expect
But how often might we expect d<c? For comparison, I used a control which replaced c by the least integer >(a+b) and prime to a and b. For the conjecture, I searched for all triples
a,b,c pairwise-coprime, a+b=c, c<10000, d<c
and for the control I searched for all triples
a,b,c, c minimally >(a+b) such that a,b,c pairwise-coprime, c<10000, d<c
and compared the results:

c = a+b 121488
c minimally > a+b 63245
Thus having c exactly equal to a+b results in significantly more hits than pairwise-coprime triples with c almost equal to a+b. This suggests the "inverse abc conjecture":
If (a,b,c) are pairwise-coprime, the probability that d<c is greater for c=a+b than for c very close to a+b.

Explanation of the above result

In fact, the result is not so surprising. It can be explained as follows.

If we choose a and b independently at random, retrying if they are not pairwise-coprime, and then calculate c=a+b,
    with probability 1, 2 divides one of (a,b,c)
    with probability 3/4, 3 divides one of (a,b,c)
    with probability 1/2, 5 divides one of (a,b,c)
    with probability 3/(p+1), p (prime) divides one of (a,b,c).
But if we choose a, b and c independently at random, retrying if they are not all pairwise-coprime,
    with probability 3/4, 2 divides one of (a,b,c)
    with probability 3/5, 3 divides one of (a,b,c)
    with probability 3/7, 5 divides one of (a,b,c)
    with probability 3/(p+2), p (prime) divides one of (a,b,c).

So three pairwise-coprime numbers such that one is the sum of the other two are more likely to include a multiple of a power of 2 than three pairwise-coprimes numbers chosen at random; and likewise for a power of 3; etc. And the more likely it is that small primes are involved, the more likely it is that d<c.

Results with a+b ≃ c

We checked the explanation above by collecting all triples a,b,c such that

The results are shown in the graph below. This is a "zoomed in" version of the graph, with a+b close to c, the difference being no more than 32.

The horizontal axis is k = c-a-b
"count" is the number of pairwise-coprime triples with that value of k
"hits" is the number of those triples with d<c

We see a very strong 2-periodicity: if k is odd, there are many fewer triples, and a lower proportion of them are hits. There are weaker periodicities for other primes.

We also see, within the range of this graph -33<k<33, that k=0, i.e. a+b=c, gives a high number of hits, and a high ratio of hits to all triples. Thus it tends to confirm the "inverse abc conjecture".

We can "zoom out" the above graph, showing all k in the range -1500 -1500 <e; k <e; 1500, and plotting only values of k divisible by 30 (which as we saw in the "zoomed in" graph above) tend to give most hits):

Again, we find that k=0, i.e. a+b=c, has a rather high proportion of hits, confirming the "inverse abc conjecture".

So what does the abc conjecture really mean?

I have two guesses about the real meaning of the abc conjecture, as studied by Mochizuki and others:

  1. The precise formula a=b+c is irrelevant. What matters is if a, b and c are all pairwise-coprime, then for every ε > 0, there exist only finitely many triples (a, b, c) such that c>rad(a*b*c)^(1+ε)
  2. The abc conjecture does not apply to the range of values we have used to create the graphs above. c < 10000. The interesting result only appears for some larger c.

Regular Maps
The graphs on this page are created by Google Chart, which is powerful, free, easy to use, and poorly documented.