A preprint of "The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits" is now available on the math arXiv. In it, Hester Graves and I show that if you assume the abc Conjecture, you can show that finitely many Cullen numbers are repunits to any base. (Sort of like it says in the title.)
Frequently Asked Questions (or so I imagine):
- What is the abc conjecture? It says that if you have integers a, b, and c with a+b=c, the product of all primes dividing a, b and c is almost never much smaller than c. For more, you can read the Wikipedia article.
- What are Cullen numbers? Cullen numbers are numbers of the form n2n+1, for a positive integer n.
- What are repunits? They are numbers all of whose digits are 1. So 111 or 1111. But! Those are base-10, and in this paper, we consider any base, so 7 is a repunit, because it is 111 in base 2. 2801 is a repunit, because it is 1111 in base 7. We don't allow 11, because every number has a base where it's 11.
- Does your result hold for s-Cullen numbers? Yes. For any given s, there are finitely many s-Cullen numbers that are repunits (assuming the abc conjecture).
- What are s-Cullen numbers? Numbers of the form nsn+1.
- When did you start using the abc conjecture? Just now! Hester had this cool idea, and it was fun to see how it worked out.