## Monday, 4 March 2013

### MONDAY, 4 MARCH 2013

Today is the $63^{rd}$ day of the year.

$63 = 7 \times 9$

$63_{\rm 10} = 111111_{\rm 2}$ which means that the binary representation of $63$ is palindromic. As can be readily inferred, any number of the form $2^{n} - 1$ will consist of a string of $n - 1$ $1$s and will therefore be palindromic, see A006995.

$3^{63} + 2 = 1,144,561,273,430,837,494,885,949,696,427 + 2$
$= 1,144,561,273,430,837,494,885,949,696,429$ which is prime, see A051783. Who would have thought that anybody would find it useful to discover all the numbers that when $3$ is raised to that power and then the result is incremented by $2$ that the answer is prime.

$63 = 3^3 + 6^2$. There are more numbers that can be written as the sum of a cube and square than I would have thought, see A055394.

Since $63 = 4 \times 2^4 - 1$ i.e it is of the form $n \times 2^n - 1$, then $63$ is a Woodall Number, see A003261.

#### 1 comment:

Coombes said...

Is there a name for primes of the form 3^n + 2, in the same way that primes of the form 2^n - 1 are called Mersenne?