Editing Highly composite number (section)

==Prime factorization==
[[File:Highly_composite_numbers.svg|thumb|250px|Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are labelled in bold and superior highly composite numbers are starred. In the [//upload.wikimedia.org/wikipedia/commons/6/60/Highly_composite_numbers.svg SVG file], hover over a bar to see its statistics.]]
Roughly speaking, for a number to be highly composite it has to have [[prime factors]] as small as possible, but not too many of the same. By the [[fundamental theorem of arithmetic]], every positive integer ''n'' has a unique prime factorization:

:<math>n = p_1^{c_1} \times p_2^{c_2} \times \cdots \times p_k^{c_k}</math>

where <math>p_1 < p_2 < \cdots < p_k</math> are prime, and the exponents <math>c_i</math> are positive integers.

Any factor of n must have the same or lesser multiplicity in each prime:

:<math>p_1^{d_1} \times p_2^{d_2} \times \cdots \times p_k^{d_k}, 0 \leq d_i \leq c_i, 0 < i \leq k</math>

So the number of divisors of ''n'' is:

:<math>d(n) = (c_1 + 1) \times (c_2 + 1) \times \cdots \times (c_k + 1).</math>

Hence, for a highly composite number ''n'',

* the ''k'' given prime numbers ''p''<sub>''i''</sub> must be precisely the first ''k'' prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than ''n'' with the same number of divisors (for instance 10&nbsp;= 2&nbsp;&times;&nbsp;5 may be replaced with 6 = 2 &times; 3; both have four divisors);
* the sequence of exponents must be non-increasing, that is <math>c_1 \geq c_2 \geq \cdots \geq c_k</math>; otherwise, by exchanging two exponents we would again get a smaller number than ''n'' with the same number of divisors (for instance 18&nbsp;=&nbsp;2<sup>1</sup>&nbsp;×&nbsp;3<sup>2</sup> may be replaced with 12&nbsp;=&nbsp;2<sup>2</sup>&nbsp;×&nbsp;3<sup>1</sup>; both have six divisors).

Also, except in two special cases ''n''&nbsp;=&nbsp;4 and ''n''&nbsp;=&nbsp;36, the last exponent ''c''<sub>''k''</sub> must equal&nbsp;1. It means that 1, 4, and 36 are the only square highly composite numbers. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of [[primorials]] or, alternatively, the smallest number for its [[prime signature]].

Note that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example, 96 = 2<sup>5</sup> × 3 satisfies the above conditions and has 12 divisors but is not highly composite since there is a smaller number (60) which has the same number of divisors.