## Thursday, 14 March 2013

### THURSDAY, 14 MARCH 2013

Today is the $73^{rd}$ day of the year.

$73$ is prime and an Emirp (see Tuesday 12 March).

$73_{10} = 1001001_2$ which means that $73$ is palindromic in binary.

There are two primitive Pythagorean triples containing $73$
$(55, 48, 73)$
$(73, 2664, 2665)$

The date today is $14^{th}$ March. If one writes the date in either the ISO format (see XKCD 1179) of $2013.3.14$ or the American format of $3.14.2013$ then it contains the sub-string $3.14$ which is the beginning of the ratio we know as $\pi$. Thus, today is known by many as Pi Day and it is used as an excuse to try and get people interested in mathematics. There is even a website.

Since $\pi$ continues infinitely without repetition or pattern then, it is proposed, any sequence of digits turns up sooner or later. There is a $\pi$ search website here that allows one to see if a particular digit sequence occurs within the $20,000,000$ digits. Out of interest the date in English format, $14022013$, occurs at position $40,231,854$.
$0$ first occurs at position $32$
$1$ first occurs at position $1$
$2$ first occurs at position $6$
$3$ first occurs at position $9$
$4$ first occurs at position $2$
$5$ first occurs at position $4$
$6$ first occurs at position $7$
$7$ first occurs at position $13$
$8$ first occurs at position $11$
$9$ first occurs at position $5$
$10$ first occurs at position $49$
...
$73$ first occurs at position $299$.
$2$ occurs for the eighth time at position $73$
The sequence of positions of $n$ within the expansion of $\pi$ which starts $32, 1, 6, 9, 2, 4, 7, 13, 11, 5, 49$ is sequence A014777 in OEIS (and $014777$ occurs at position $2,025,745$).
My birthday in $ddmmyyyy$ format occurs at position $198,662,921$.

$73$ is the fourth member of sequence A023001,  whose first nine members are:
$0, 1, 9, 73, 585, 4681, 37449, 299593, 2396745$
The definition of this sequence is $a(n) = \frac {8^n - 1}{7}$. What is fascinating about this is the implication that every power of $8$ less one is divisible by seven. Can anyone find a proof for that?