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.
So what does the abc conjecture say? According to Wikipedia:
According to Nature (October 7, 2015) it says
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".
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:
a | b | c | d |
---|---|---|---|
1 | 8 | 9 | 6 |
1 | 63 | 64 | 42 |
1 | 80 | 81 | 30 |
32 | 49 | 81 | 42 |
3 | 125 | 128 | 30 |
13 | 243 | 256 | 78 |
1 | 2400 | 2401 | 210 |
So I wanted to test the conjecture:
d/c < 1 | d/c < 0.5 | d/c < 0.2 | |
---|---|---|---|
c = a+b | 121 | 48 | 8 |
c minimally > a+b | 63 | 24 | 5 |
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.
We checked the explanation above by collecting all triples a,b,c such that
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".
I have two guesses about the real meaning of the abc conjecture, as studied by Mochizuki and others:
Weddslist
Regular Maps
The graphs on this page are created by Google Chart,
which is powerful, free, easy to use, and poorly documented.