Definition:Solid Number

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A solid number is the product of three (natural) numbers.

In the words of Euclid:

And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.

(The Elements: Book $\text{VII}$: Definition $17$)


The side of a solid number is one of the (natural) numbers which are its divisors.


A example of a solid number is $12 = 2 \times 2 \times 3$.

Its divisors $2$ and $3$ are its sides.

Similar Numbers

Let $m$ and $n$ be solid numbers.


$m = p_1 \times p_2 \times p_3$ where $p_1 \le p_2 \le p_3$
$n = q_1 \times q_2 \times q_3$ where $q_1 \le q_2 \le q_3$

Then $m$ and $n$ are similar if and only if:

$p_1 : q_1 = p_2 : q_2 = p_3 : q_3$

Also see