Monday 26 February 2018

Fermat Primes and Brazilian Numbers

Today I turned \( 25156 \) days old and one of the entries, AO63799, in the OEIS (Online Encyclopaedia of Integer Sequences) for this number states that: \( 25156 \) belongs to the set of numbers \( n \) such that n+3, n+5, n+17, n+257, n+65537 are all primes.

At first the numbers \(3, 5, 17, 257 \text{ and } 65537 \) appeared quite arbitrary but helpfully the comment is made that these numbers are the known Fermat primes. These are primes of the form:$$ 2^{2^k} + 1, \text{ for some k } >= 0 $$ It is conjectured that there are only five values of \( k \) that produce such primes, namely \( 0, 1, 2, 3 \text{ and } 4 \) corresponding to \( 3, 5, 17, 257 \text{ and } 65537 \). It has been confirmed that values of k such that \( 5 \leq k \leq 32 \) produce composite numbers.

Of course, these primes are not be confused with the Mersenne primes that are of the form \( 2^k-1 \) and that are probably infinite in number.

In the comments for OEIS AO63799, it's also stated that no Fermat prime is a Brazilian number which of course immediately prompted me to find what defined a Brazilian number. 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 \( 2p=2(p-1)+2 \) is written \( 22 \) in base \(p-1 \text{ if } p-1>2 \), 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(14-1)+2 and thus can be represented as \(22_{13} \).

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