When my first journal article was published in 1995, putting reprints on my personal web site seemed very advanced compared to keeping them on a shelf in my office.

Times have changed, however, and arXiv.org seems like a more permanent repository for the reprints than pseudoprime.com. And, frankly, more permanent than some of the journal web sites.

So you can now see all of my reprints (and preprint!) at this arXiv link.

## Wednesday, March 20, 2019

## Wednesday, March 13, 2019

### 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+b

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 10

We have actually computed all Brazilian Sophie Germain primes up to 10

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

What does this mean? A Brazilian prime is a prime number of the form 1+b+b

^{2}+...+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 10

^{44}. 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 10^{52}.We have actually computed all Brazilian Sophie Germain primes up to 10

^{46}(there are 104,890,302 of them) and 10^{48}(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.

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