Brazilian Primes

There's only one reference that I've made to Brazilian primes in this blog and that was in a post titled Fermat Primes and Brazilian Numbers on February 26th 2018 which is now over five years ago. I was reminded of them once again when a number associated with my diurnal age, 27061, was identified as a Brazilian prime. I wrote the following about Brazilian numbers in that earlier post:

These numbers are listed in OEIS A125134 and defined as:

Numbers \( n \) such that there is a natural number \( b \) with \( 1 < b < n-1 \) such that the representation of \( n \) in base \( b \) has all equal digits.

All even numbers \( \geq \) 8 are Brazilian numbers because: $$ \begin{align} 2p&=2(p-1)+2 \\&= 22 \end{align}$$in base \(p-1\) if \(p-1>2 \) and that is true if \(p \geq 4 \). 

The odd Brazilian numbers are listed in OEIS A257521 and are fairly common, with some being prime numbers:

7, 13, 15, 21, 27, 31, 33, 35, 39, 43, 45, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 127, 129, 133, 135, 141, 143, 145, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 183, 185, 187, 189, 195, ...

As an example, take 27 in the above sequence which can be expressed as \(33_8 \). As for the even numbers, take a number like 28. It can be written as 2 x (14-1) + 2 and thus can be represented as \(22_{13} \).

Brazilian primes are simply Brazilian numbers that are prime. These constitute OEIS A085104:

A085104

Primes of the form \(1 + n + n^2 + n^3 + ... + n^k \) where \(n > 1\) and \( k > 1\).

For example, in the case of 27061 we have: $$ \begin{align} 27061& = 1+164+164^2 \\ &= 164^2+164+1\\&=111_{164} \end{align}$$These primes can be generated using this permalink. The initial members of the sequence are:

7, 13, 31, 31, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023, 20593, 21757, 22621, 22651, 23563, 24181, 26083, 26407, 27061, 28057, 28393, 30103, 30941, 31153, 35533, 35911, 37057, 37831, 41413, 42643, 43891, 46441, 47743, 53593, 55933, 55987, 60271, 60763, 71023, 74257, 77563, 78121, 82657, 83233, 84391, 86143, 88741, 95791, 98911

The OEIS comments to this sequence are informative:

The number of terms \(k+1\) is always an odd prime, but this is not enough to guarantee a prime, for example 111 = 1 + 10 + 100 = 3 x 37.

The inverses of the Brazilian primes form a convergent series; the sum is slightly larger than 0.33.

It is not known whether there are infinitely many Brazilian primes. 

Brazilian primes can be written in the form:

$$ \dfrac{(n^p - 1)}{(n - 1}\\ \text{ where } p \text{ is an odd prime and } n > 1$$

The number of terms less than \(10^n\) are 1, 5, 14, 34, 83, 205, 542, 1445, 3880, 10831, 30699, 88285, ...

Brazilian primes fall into two classes:

• when \(n\) is prime, we get sequence OEIS A023195 except 3 which is not Brazilian,
• when \(n\) is composite, we get sequence OEIS A285017.

The conjecture that "No Sophie Germain prime is Brazilian (prime)"  is false because:$$ \begin{align} a(856) &= 28792661\\ &= 1 + 73 + 73^2 + 73^3 + 73^4 \\&= (11111)_{73} \end{align} $$and 28792661 is the 141385-th Sophie Germain prime.