= Perfect number
{title2=$s(n)=n\iff\sigma(n)=2n$}
{wiki}
A <perfect number> is a <positive integer> equal to the <sum> of all its positive <proper divisors>. The divisor itself is excluded: including it gives the equivalent condition $\sigma(n)=2n$ for the <sum-of-divisors function>. The smallest examples are \b[$6=1+2+3$] and \b[$28=1+2+4+7+14$].
For comparison, $8$ has <proper divisors> $1,2,4$, whose <sum> is $7$: it is a <deficient number>. The <proper divisors> of $12$ are $1,2,3,4,6$, whose <sum> is $16$: it is an <abundant number>. The <proper-divisor sum> classifies these examples with the same rule, and gives $s(1)=0$, so one is not a <perfect number>.
\Image[/perfect-number-divisor-balances.png]
{title=Proper divisors balance a perfect number, fall short of a deficient number, or exceed an abundant number}
{description=Original exact <proper-divisor sums> for $6,8,12,28$. Each bar is normalized by its candidate number, and its segments are labelled with the actual <proper divisors>. The dashed line at one marks equality with the candidate. These are finite arithmetic calculations, not empirical data.}
The interesting question is how such balances can be constructed without testing every smaller <integer> as a possible <divisor>. The <Euclid-Euler theorem> answers it for <even integers>: a <Mersenne prime> $2^p-1$ gives $2^{p-1}(2^p-1)$, and every even <perfect number> arises this way. For $p=2,3,5,7$, this produces $6,28,496,8128$. The connection links divisor sums to <primality testing> for a particular family, while keeping the necessary hypotheses visible.<Perfect numbers and Mersenne primes by Keith Conrad>[§1, Theorem 1.1 and Example 1.2, pp. 1–2]{ref}
<Euclid>'s <Elements> already gives the construction in Book IX, Proposition 36: add a succession of <powers of two>, and if their <sum> is <prime>, multiply by the final <power of two>. The modern <finite geometric series> $1+2+\cdots+2^{p-1}=2^p-1$ makes the same pattern transparent. This original statement supplies the forward construction, rather than the converse classification of all even <perfect numbers>.<Euclid's Elements by Euclid/Book IX, Proposition 36>[Book IX, Proposition 36: statement and concluding proof]{ref}
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