Publication of "No New Goormaghtigh Primes up to 10^700"

 

My paper, "No New Goormaghtigh Primes up to 10⁷⁰⁰" appeared in Integers last year.

This was a super-fun paper to write. I had not heard of the Goormaghtigh Conjecture until a virtual talk I attended during the pandemic. I stopped listening 10 minute in when the speaker moved on to generalizations and had a hard time stopping thinking about the topic.

The conjecture is that the only two numbers which are repunits (all 1s) in two different bases are 31 and 8191. I came up with the idea of restricting to prime solutions and was able to show that those are the only two such numbers up to 10⁷⁰⁰.

Preprint of "Only finitely many s-Cullen numbers are repunits for a fixed s≥2" on the math arXiv

A preprint of my paper with Michael Filaseta and Hester Graves, "Only finitely many s-Cullen numbers are repunits for a fixed s≥2" has appeared on the math arXiv. This is a sequel to an earlier paper with Hester, which proved that there were only finitely many solutions for all s, subject to the abc conjecture.

This proof is unconditional. I am now working on computations to extend 65536 to a much larger number -- it should be possible to get it into the millions.

Preprint of "Finding a Widely Digitally Delicate Prime" available on the math arXiv

In September, I posted a preprint of "Finding a Widely Digitally Delicate Prime" on the math arXiv. It may not end up published in this form, but because it described the only known example of such a prime, and how I found it, I wanted to have it documented.

"The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits" Appears in the Journal of Integer Sequences

 

Catching up from earlier this year, my paper with Hester Graves, "The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits" appeared in the Journal of Integer Sequences.

This is my ninth journal article published.

Preprint of "The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits" on the arXiv

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):

1. 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.
2. What are Cullen numbers? Cullen numbers are numbers of the form n2ⁿ+1, for a positive integer n.
3. 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.
4. 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).
5. What are s-Cullen numbers? Numbers of the form nsⁿ+1.
6. 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.
   

Talk on Brazilian Primes at SERMON 2019

Here are the slides of the talk I am giving today at the SERMON 2019 conference. It is similar to previous talks I have given on primes that are values of cyclotomic polynomials, but with more emphasis on Brazilian primes, in particular Brazilian Sophie Germain primes.

Preprint of "Brazilian Primes Which Are Also Sophie Germain Primes"

A preprint of "Brazilian Primes Which Are Also Sophie Germain Primes" is now available on the math arXiv. In it, Hester Graves and I disprove a conjecture from Schott's 2010 paper on Brazilian primes, namely that no Brazilian primes are also Sophie Germain primes.

What does this mean? A Brazilian prime is a prime number of the form 1+b+b²+...+b^k-1, i.e. a prime whose digits are all 1 when written in base b. (To avoid silliness, you need k>2 and b>1.) For this reason, they are sometimes called "prime repunits".

A Sophie Germain prime is a prime p such that 2*p+1 is also prime.

If you just start computing Brazilian primes, most of them will be of length 3. We show that the length of a Brazilian Sophie Germain prime has to be a prime congruent to 2 mod 3, i.e. 5, 11, 17, 23, etc.

In the paper, we computed all Brazilian Sophie Germain primes up to 10⁴⁴. There are 38,031,404 of them, all but 12 of them of length 5. The 12 exceptions are all of length 11. The smallest one of length 17 is 41969813142886369903423014255641324842178685773056721, which is bigger than 10⁵².

We have actually computed all Brazilian Sophie Germain primes up to 10⁴⁶ (there are 104,890,302 of them) and 10⁴⁸ (we haven't counted them up yet). A later version of the preprint will reflect that.

Submission of the sequence of Brazilian Sophie Germain primes is in progress. A later version of the preprint will also reflect that.