# 729

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## Number

$729$ (seven hundred and twenty-nine) is:

$3^6$

$1 \, 000 \, 000$ in base $3$

The $1$st solution to the approximate Fermat equation $x^3 = y^3 + z^3 \pm 1$:
$9^3 = 6^3 + 8^3 + 1$

The $1$st cube which can be expressed as the sum of $5$ positive cubes:
$729 = 1^3 + 3^3 + 4^3 + 5^3 + 8^3$

The larger of the $1$st pair of Smith brothers:
$7 + 2 + 8 = 2 + 2 + 2 + 7 + 1 + 3 = 17$, $7 + 2 + 9 = 3 + 3 + 3 + 3 + 3 + 3 = 18$

The $2$nd cube which can be expressed as the sum of $3$ positive cubes:
$729 = 1^3 + 6^3 + 8^3$

The $3$rd $6$th power after $1$, $64$:
$729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3$

The number of different commutative binary operations that can be applied to a set with $3$ elements

The $3$rd power of $9$ after $(1)$, $9$, $81$:
$729 = 9^3$

The $6$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$:
$729 = 3^6$

The $9$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$:
$729 = 9 \times 9 \times 9$

The $27$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $361$, $400$, $441$, $484$, $529$, $576$, $625$, $676$:
$729 = 27 \times 27$

The $38$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$, $663$, $666$, $690$, $706$, $728$:
$7 + 2 + 9 = 3 + 3 + 3 + 3 + 3 + 3 = 18$

The $46$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $400$, $432$, $441$, $484$, $500$, $512$, $529$, $576$, $625$, $648$, $675$, $676$:
$729 = 3^6$

## Historical Note

$729$ was particularly significant to the Pythagoreans, as it is $27^2$ as well as being $9^3$.

The philosopher Plato rejected the well-known length of the year as being (approximately) $365 \frac 1 4$ in favour of $364 \frac 1 4$, as the latter is $729$, that is $9^3$, days and nights.

This was on the grounds that:

... if one were to express the extent of the interval by which the tyrant is parted from the king in respect of true pleasure he will find on completion of the multiplication that he lives $729$ times as happily and that the tyrant's life is more painful by the same distance.
-- Plato's Republic: $588$

Various other interpretations of this passage have been suggested.

Plato combined the sequences of squares and cubes and their roots from $1$ to $3$ to get:

$1 + 2 + 3 + 4 + 8 + 9 = 27$

Charles Albert Browne, Jr. pointed out that the $27 \times 27$ magic square, being filled with the numbers from $1$ to $729$, has $365$ in the centre cell.