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Marin Mersenne was a mathematician and scientific correspondent whose network helped Pierre de Fermat, René Descartes and other researchers exchange questions and results. He studied number theory and the acoustics of vibrations of mechanical strings.^[ref] A Mersenne number has the form $2^p-1$; for it to be a Mersenne prime, $p$ must be a prime number, although that condition is not sufficient: $2^{11}-1=23\cdot 89$. For an ideal stretched mechanical string, the fundamental frequency is $f=(2L{)}^{-1}\sqrt{T/\mu }$, relating the audible pitch to the length $L$, tension $T$ and mass per unit length $\mu$.

A Mersenne number has the form $M_k=2^k-1$ for a positive integer $k$. Its binary representation consists of $k$ ones. For example, $M_4=15$ is composite, while $M_5=31$ is prime. If $k=ab$ with $a,b>1$, the finite geometric series factorizationhas two factors greater than one. Consequently a Mersenne prime requires a prime exponent, but $M_{11}=23\cdot 89$ shows that this condition alone is insufficient.

A Mersenne prime is a Mersenne number that is a prime number. It is the prime factor required by the Euclid-Euler theorem; $3,7,31,127$ generate $6,28,496,8128$. The family is named after Marin Mersenne.

The passage belongs to the English reception of Euclid's number theory. Henry Billingsley's The Elements of Geometrie by Henry Billingsley, published in 1570, brought the Elements into English with extensive explanatory material. The original title page advertises translation and annotations; the extract's worked examples show what explanatory prose could add to a compact definition. Henry Billingsley's numbering calls this Definition 23 of Book VII; the familiar modern numbering is VII, Definition 22. The difference is editorial, not a difference in the perfect number being defined.^[ref][ref]

Here a number's parts are its positive proper divisors, rather than arbitrary pieces of a sum or all smaller integers. In modern notation the definition iswhere $\sigma$ is the sum-of-divisors function. Thus $12$ is not a perfect number: its positive proper divisors total $1+2+3+4+6=16$. It is essential to exclude $n$ itself: including it would double the required sum. The pairing of a divisor $d$ with the fraction $1/(n/d)$ of the whole explains the extract's language of halves, thirds and so on. Its examples are complete checks because they identify every positive proper divisor, not merely some convenient decomposition.

The larger mathematical context is the construction in Book IX, Proposition 36. If $q=2^p-1$ is a prime number, then $N=2^{p-1}q$ is a perfect number. To reconstruct the argument, prime factorization gives all its divisors as $2^j$ and $q{2}^j$, with $0\le j\le p-1$. A finite geometric series therefore givesThe choices $p=2,3,5,7$ produce $6,28,496,8128$. The primality assumption matters: choosing $p=4$ produces $120$, whose sum-of-divisors function is $360$, not $240$. The construction gives a reason for the examples and connects perfect numbers with Mersenne primes.^[ref]

The extract itself supplies a definition and examples, not a classification or a mathematical proof that such numbers are rare. In particular, the converse that every even perfect number has this form belongs to the later Euclid-Euler theorem; it should not be read back into this passage. The historically significant feature is the transmission of deductive arithmetic through vernacular explanation: a book called geometry also teaches the divisibility theory needed for a general construction.

Using the assumed square-free polynomial property, factorinto distinct monic polynomials that are irreducible polynomials, with $n_j=\mathrm{deg}{f}_j$. Part (b)(iii) identifies $T={\mathbb{F}}_2[X]/(X^p-1)$ with ${\prod }_j{K}_j$, where $|K_j|=2^{n_j}$.

By the first isomorphism theorem for rings, the image $S$ from part (c)(i) is $T/J$, where $J$ is the kernel of a ring homomorphism. By part (a)(ii), $J={\prod }_j{J}_j$ for ideals $J_j\subseteq K_j$. A field has only the zero ideal and the whole field as ideals, so each $K_j/J_j$ is either $K_j$ or the zero ring. Consequently, for a subset $D\subseteq \{1,\dots ,r\}$,This is the quotient of a finite product of fields principle. The subset is nonempty because $|U(S)|=p>2$, whereas the zero ring has only one unit. By part (a)(i),Each factor is a positive integer. Since $p$ is a prime number, exactly one factor is $p$ and all remaining factors are one. Thusso $p$ is a Mersenne prime. If $n=ab$ with $a,b>1$, then $2^a-1$ would be a proper divisor of $2^{ab}-1$; thus $n$ is itself a prime number, although only the displayed form is needed.

The assumed square-free polynomial property can also be checked directly: in characteristic of a ring two, the formal derivative of $X^p-1$ is $X^{p-1}$ because $p$ is odd. These two polynomials have greatest common divisor one, since $X$ does not divide $X^p-1$, so no repeated irreducible polynomial factor occurs.

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 $6=1+2+3$ and $28=1+2+4+7+14$.

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.

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.^[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.^[ref]

The prime exponents $2,3,5,7,13$ give Mersenne primes and therefore perfect numbers. But $p=11$ gives $2^{11}-1=2047=23\cdot 89$. The resulting candidate $N=2^{10}\cdot 2047=2096128$ hasso $s(N)=2325392>N$. It is an abundant number, not a perfect number. Checking only that the exponent is prime would miss this obstruction.