## Saturday, 2 March 2013

### SATURDAY, 2 MARCH 2013

Today is the $61^{st}$ day of the year.

$61$ is prime.

$2^{61} - 1 = 2,305,843,009,213,693,951$ which is also prime, therefore $61$ is a Mersenne exponent, see A000043.

$61 = 4.15 + 1$. All primes of that are 1 Modulo 4 can be expressed as the sum of two squares, see A002144. In this case $61 = 6^2 + 5^2 = 36 + 25$

Consider the Fibonacci like sequence that starts with the digits of $61$.
$a(0) = 6$
$a(1) = 1$
$a(2) + a(1) + a(0) = 6 + 1 = 7$
$a(3) = 7 + 1 = 8$
$a(4) = 8 + 7 = 15$
$a(5) = 15 + 8 = 23$
$a(6) = 23 + 15 = 38$
$a(7) = 38 + 23 = 61$
Since $61$ appears in a sequence that starts with its own digits then it is a Keith Number, see A007629.