Open and solved problems about the abc conjecture
What is the abc conjecture | Open and solved problems | Consequences | The algorithm
If you create a list of abc triples, you might notice the same c appears in multiple triples. Some appear only once, others two times, and others even more. We know for every n there is a c that appears in n triples. But what is the smallest c that occurs in 2 triples? what is the smallest that appears in 3 triples? And 4? The same is known for b, but again we do not know what the smallest b is that occurs in n triples.
We know that every a appears in infinitely many triples.
Which values appear for b - a, when a, b and c are an abc triple? Do some values never appear? Do some values appear more often than others?
There are infinitely many triples, but how much are there with c < n? And for how many triples is b less than 2a?
When you have an abc triple, a, b or c is divisible by 2. Is there for every n an abc triple so a, b and c are not divisible by 3, 4, 5, ..., n? What is the smallest triple when n = 13?
Two triples are called a twin when both triples have the same c and the same radical. In this case, the triples also have the same quality. Can we find infinitely many of these twins?
If a + b = c, a and b are coprime and rad(abc) < c we have an abc triple. We can also look at non-abc triples where the radical may be bigger than c, so q < 1. Can we construct a sequence so q converges to 1/3?
Wim Voorn recently solved this problem by considering discovering there must be infinitely many squarefree triples of the form x, x+1, 2x+1.