# 73

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## Number

$73$ (seventy-three) is:

The $21$st prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$, $67$, $71$

The only Sheldon prime.

The $1$st of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$

The $2$nd of the $8$th pair of twin primes, with $71$

The $1$st of the $2$nd ordered quadruple of consecutive integers that have sigma values which are strictly increasing:
$\sigma \left({73}\right) = 74$, $\sigma \left({74}\right) = 114$, $\sigma \left({75}\right) = 124$, $\sigma \left({76}\right) = 140$

The lower end of the $5$th record-breaking gap between twin primes:
$101 - 73 = 28$

The $6$th emirp after $13$, $17$, $31$, $37$, $71$

The $8$th two-sided prime after $2$, $3$, $5$, $7$, $23$, $37$, $53$:
$73$, $7$, $3$ are prime

The $9$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$

The $11$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$

The $12$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$, $59$, $67$, $71$ such that the Mersenne number $2^p - 1$ is composite

The $14$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$, $67$ such that no factorial of an integer can end with $n$ zeroes

The $17$th lucky number:
$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $\ldots$

Every positive integer can be expressed as the sum of at most $73$ $6$th powers

In the smallest equilateral triangle with sides of integer length ($112$) which contains a point which is an integer distance from each vertex, the distance from that point to its furthest vertex (the other two being $57$ and $65$)

## The Sheldon Cooper Exposition

$73$ is the best number:
$73$ is the $21$st prime number.
Its mirror $37$ is the $12$th prime number.
Its mirror $21$ is the product of multiplying, hang on to your hats, $7$ by $3$.
Eh? Eh? Did I lie?
We get it. $73$ is the Chuck Norris of numbers.
Chuck Norris wishes.
In binary, $73$ is a palindrome: $1,001,001$, which backwards is $1,001,001$, exactly the same.
All Chuck Norris backwards gets you is Sirron Kcuhc!

It may be worth adding the following observation about the Hilbert-Waring Theorem:

Every positive integer can be expressed as the sum of at most $37$ positive $5$th powers.

and:

Every positive integer can be expressed as the sum of at most $73$ (positive) $6$th powers.