Nvmeri primi et theorema fvndamentale arithmeticae

Svmmarivm

Haec lectio nvmeros primos tractat, atomos mvltiplicationis nvmerorvm natvralivm. Demonstrat omnem nvmervm natvralem maiorem qvam $1$ divisorem primvm habere, cribrvm Eratosthenis exponit et cavsam cvr satis sit vsqve ad radicem qvadratam cribrare, et cvm Evclide ostendit nvmeros primos infinitos esse. Deinde lemma Evclidis demonstrat, secvndvm qvod nvmervs primvs qvi prodvctvm dividit aliqvem eivs factorem dividit, et eo adhibito theorema fvndamentale arithmeticae: omnis nvmervs natvralis maior qvam $1$ vno tantvm modo vt prodvctvm nvmerorvm primorvm scribitvr. Ex factorizatione dedvcvntvr nvmervs divisorvm nvmeri natvralis, characterismvs qvadratorvm perfectorvm et irrationalitas radicis qvadratae omnis nvmeri natvralis qvi qvadratvm perfectvm non sit.

Proposita discendi

Nvmeros primos a compositis distingvere et demonstrare omnem nvmervm natvralem maiorem qvam $1$ divisorem primvm habere.
Cribrvm Eratosthenis adhibere et explicare cvr satis sit mvltipla eorvm nvmerorvm primorvm delere qvi radicem qvadratam finis non excedvnt.
Infinitatem nvmerorvm primorvm, lemma Evclidis et exsistentiam et vnicitatem factorizationis in nvmeros primos demonstrare.
Factorizatione vti ad divisores nvmerandos et envmerandos, ad qvadrata perfecta agnoscenda et ad irrationalitatem radicvm qvadratarvm demonstrandam.

Nvmeri primi et nvmeri compositi

In lectione de divisibilitate et divisione cvm residvo visvm est omnem nvmervm natvralem $n$ per $1$ et per se ipsvm divisibilem esse. Qvidam nvmeri natvrales nvllos alios divisores admittvnt praeter hos dvos; ii svnt partes elementares ex qvibvs, per mvltiplicationem, omnes ceteri componvntvr.

Definitio (nvmervs primvs). Nvmervs natvralis $p$ est primvs si $p > 1$ et soli eivs divisores svnt $1$ et $p$ . Nvmervs natvralis $n > 1$ qvi primvs non est compositvs appellatvr: scribitvr vt $n = ab$ , cvm $1 < a < n$ et $1 < b < n$ . Nvmervs $1$ neqve primvs neqve compositvs est.

Nvmeri primi minores qvam $50$ svnt hi qvindecim:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Nvmervs $2$ solvs nvmervs primvs par est, qvia omnis alivs nvmervs par divisorem $2$ habet, a $1$ et a se ipso diversvm. Exclvsio nvmeri $1$ non est arbitrivm definitionis: si $1$ primvs esset, factorizatio in nvmeros primos vnica esse desineret, qvia $6 = 2 \cdot 3 = 1 \cdot 2 \cdot 3$ , et theorema praecipvvm hvivs lectionis cvm exceptione envntiandvm esset.

Principivm bonae ordinationis

Demonstrationes hvivs lectionis in qvadam forma principii indvctionis nitvntvr qvae ad ratiocinandvm de divisoribvs praecipve commoda est: in principio bonae ordinationis, secvndvm qvod omnis copia non vacva nvmerorvm natvralivm elementvm minimvm habet. Ex indvctione forti dedvctvm est in lectione de svmmationibvs, progressionibvs et indvctione, et hic adhibetvr demonstratione eivs non repetita.

Omnis nvmervs natvralis maior qvam 1 divisorem primvm habet

Primvm lectionis effatvm confirmat nvllvm nvmervm natvralem maiorem qvam $1$ nvmeros primos effvgere: semper est aliqvis qvi evm dividit, et inter minimos divisores invenitvr.

PropositioMinimvs divisor maior qvam 1 primvs est

Omnis nvmervs natvralis $n > 1$ divisorem primvm habet: minimvs divisor nvmeri $n$ maior qvam $1$ primvs est. Si praeterea $n$ compositvs est, minimvs eivs divisor primvs $p$ condicionem $p^2 \leq n$ explet. Figvra in recta divisores nvmeri $n = 91 = 7 \cdot 13$ notat.

Demonstratio

$\dato{D}{D = \{ d \in \mathbb{N} : d \mid n \wedge d > 1 \}}, \quad n \in D, \quad m = \min D$
Copia divisorvm nvmeri $n$ maiorvm qvam $1$ vacva non est, qvia ipsvm $n$ continet; per principivm bonae ordinationis, elementvm minimvm $m$ habet.
$a \mid m \wedge m \mid n \rightarrow \resaltar{a \in D}, \quad \resaltar{a < m}$
Si $m$ compositvs esset, $m = ab$ cvm $1 < a < m$ , factor $a$ divideret $m$ et, per transitivitatem divisibilitatis, $n$ : esset elementvm copiae $D$ minvs qvam $m$ , contra electionem nvmeri $m$ . Ergo $m$ primvs est.
$\dato{k}{n = pk, \quad k \in D}$
Sit nvnc $n$ compositvs, et sit $p$ minimvs eivs divisor primvs, qvi est elementvm minimvm copiae $D$ . Scribimvs $n = pk$ ; qvia $n$ primvs non est, $k \neq 1$ , et $k$ qvoqve dividit $n$ .
$p^2 = p \cdot p \leq \resaltar{p \cdot k} = n$
Propter minimalitatem nvmeri $p$ , habetvr $k \geq p$ , et per monotoniam mvltiplicationis margo obtinetvr.

Ergo, vt diivdicetvr vtrvm nvmervs natvralis $n$ primvs sit, satis est nvmeros primos $p$ cvm $p^2 \leq n$ experiri: si nvllvs evm dividit, $n$ primvs est. Exempli gratia, pro $221$ satis est nvmeros primos vsqve ad $13$ experiri, qvia $17^2 = 289 > 221$ : non est divisibilis per $2$ , $3$ , $5$ , $7$ neqve $11$ , sed per $13$ divisibilis est, et $221 = 13 \cdot 17$ . Contra, $211$ per nvllvm ex nvmeris primis $2$ , $3$ , $5$ , $7$ , $11$ et $13$ divisibilis est, et qvia $17^2 = 289 > 211$ , nvmervs $211$ primvs est.

In reliqva parte hvivs lectionis

Cribrvm Eratosthenis
Nvmeri primi infiniti svnt
Lemma Evclidis
Theorema fvndamentale arithmeticae
Conseqventiae factorizationis
Divisores nvmeri natvralis
Qvadrata perfecta
Radices qvadratae irrationales
Problemata solvta
Clavis cryptographica infirma
Minimvs nvmervs natvralis cvm dvodecim divisoribvs
Centvm nvmeri compositi consecvtivi
Infiniti nvmeri primi qvi residvvm 3 relinqvvnt per 4 divisi
Nvmeri perfecti Evclidis
Problemata proposita
Nvmeri primi formae $n^4 + 4$
Ostia andronis
Divisor commvnis qvadratorvm nvmerorvm primorvm
Dvplvm qvadratvm et triplvm cvbvs
Nvmeri primi Mersenniani