666

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Number

$666$ (six hundred and sixty-six) is:

$2 \times 3^2 \times 37$


The $3$rd and last after $55$, $66$ of the $3$ repdigit numbers which are also triangular.


The total of all the entries in a magic square of order $6$, after $1$, $(10)$, $45$, $136$, $325$:
$666 = \ds \sum_{k \mathop = 1}^{6^2} k = \dfrac {6^2 \paren {6^2 + 1} } 2$


The $7$th after $4$, $13$, $38$, $87$, $208$, $377$ in the sequence formed by adding the squares of the first $n$ primes:
$666 = \ds \sum_{i \mathop = 1}^7 {p_i}^2 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2$


The $8$th positive integer after $1$, $24$, $26$, $87$, $168$, $388$, $594$ whose Euler $\phi$ value is equal to the product of its digits:
$\map \phi {666} = 216 = 6 \times 6 \times 6$


The $9$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$


The $34$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$, $663$:
$6 + 6 + 6 = 2 + 3 + 3 + 3 + 7 = 18$


The $36$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$:
$666 = \ds \sum_{k \mathop = 1}^{36} k = \dfrac {36 \times \paren {36 + 1} } 2$


The basis of an approximation to the Golden Ratio correct to $10$ decimal places:
$\phi \approx -2 \map \sin {666} = -1.61803 \, 39887 \, 5 \ldots$


Arithmetic Functions on $666$

\(\ds \map \phi { 666 }\) \(=\) \(\ds 216\) $\phi$ of $666$


Also see


Historical Note

$666$ is the famous Number of the Beast of the Book of the Revelation:

Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.
-- The Book of Revelation, chapter 13, verse 18

Hence its reputation as a number traditionally associated with the occult.


Less well known is the fact that the number also appears in the Book of Kings:

Now the weight of gold that came to Solomon in one year was six hundred threescore and six talents of gold.
-- The First Book of Kings, chapter 10, verse 14


It is also the number that can be formed by all the Roman numerals less than $\mathrm M$ used once each:

$666 = \mathrm {DCLXVI}$


It is possible that $666$ was merely chosen as a representative example of some large unspecified number, in the same way that $101$ or $1001$ is frequently so used in contemporary milieux.

However, its other properties make it far more interesting than that.


Sources

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