2-N\303\272meros Primos & Fatora\303\247\303\243oEstudaremos os conceitos de n\303\272meros primos e divis\303\255veis, entender como encontr\303\241-los, fator\303\241-los e sobre a infinidade de n\303\272meros primos existentes. Tamb\303\251m sobre os n\303\272meros ditos square-free .

restart; with(plots):

Warning, the name changecoords has been redefined

Primos := proc( N ) local n, m, M; n := min(14, floor( sqrt(N)*1.5 )); m := ceil( N/n); print(cat(`O primeiro`, n*m, ` primos : `)); print(` `); array( [seq( [ seq( ithprime( (j-1)*n+i), j = 1..n ) ], i = 1..m) ]); end proc: Crivo de Erist\303\263tenes := proc( N ) local n, m, A,M, p,i,j,k, K; n := min(20, floor( sqrt(N)*1.5 )); m := ceil( N/n); M := n*m; A:= array( [seq( [ seq( (i-1)*n+j+1, j = 1..n ) ], i = 1..m) ]): print(cat(`Abaixo est\303\243o os inteiros a partir do 2 at\303\251 `, M+1)); print(` `); print(A); print(` `);print(` `);print(` `); K := 0; for K while ( ithprime(K+1) < evalf(sqrt(M))) do end do; for k from 1 to K do p := ithprime(k); for i from 1 to m do for j from 1 to n do if(A[i,j]<>p) then if( irem(A[i,j],p)=0) then A[i,j]:=` ` fi; fi; od;od; print(cat(`Abaixo est\303\243o os inteiros a partir do 2 at\303\251 `, M+1, ` depois de remover os m\303\272ltiplos de `,p,` (exceto o pr\303\263prio `,p,` )`)); print(` `); print(A);print(` `);print(` `); od; print(` `); print(`O Crivo de Erat\303\263stenes.Diz que todo o n\303\272mero deixado no crivo \303\251 um n\303\272mero primo`); print(`porque n\303\263s removemos todos os n\303\272meros compostos !!`); end proc: ;

Error, missing operator or `;`

LargestPrime := proc(p) local n,k, P, N,NS,product; n := 0; for n while ( ithprime(n) < p) do end do; P:= ithprime(n); print(cat(`Se `, P, ` (O `,n,\302\272 `primo) \303\251 o maior n\303\272mero primo, ent\303\243o os primos podem ser :`)); product := 1; for k from 1 to n do product := product* ithprime(k); print(ithprime(k)); od; print(` `); print(cat(`O produtos desses n\303\272meros primos conhecidos \303\251 ` , product)); N := product +1 ; print(cat(`Se adicionarmos um a esse produto, n\303\263s temos o numero `, N)); NS := numtheory[factorset](N); if ( nops(NS) = 1) then print(cat(N,` \303\251 o novo n\303\272mero primo sozinho!`)); else print(cat(N,` n\303\243o \303\251 primo, mas isto possui novos primos.`)); print(N = ifactor(N)); fi; end proc: ;

Error, missing operator or `;`

FindSquare := proc(N) local n,k,FS, square; if( numtheory[issqrfree](N) ) then print(cat(N,` n\303\243o tem raiz exata!`)); else FS := numtheory[factorset](N); square := 1; n:= nops(FS); for k from 1 to n do if( irem(N, FS[k]^2) = 0) then square := square*FS[k]^2; fi; od; print(cat(square, ` \303\251 o melhor quadrado que divide `,N)); fi; end proc:

Considerando os n\303\272meros naturais diferentes de um, existem dois tipos de n\303\272meros : - n\303\272meros primos - s\303\243o os que podem ser divididos apenas por 1 ou por si pr\303\263prio. - n\303\272meros divis\303\255veis - os quais tamb\303\251m podem ser fatorados por outros n\303\272meros. `N\303\272meros primos: `; 7: %=ifactor(%); 31: %=ifactor(%); 173: %=ifactor(%);

NiMlM058anl8anZ8aHZtZXJvc35wcmltb3M6fkc=NiMvIiIoLSUhRzYjIiIoNiMvIiNKLSUhRzYjIiNKNiMvIiR0Ii0lIUc2IyIkdCI=

`N\303\272meros compostos: `; 6: %=ifactor(%); 35: %=ifactor(%); 221: %=ifactor(%);

NiMlNk58anl8anZ8aHZtZXJvc35jb21wb3N0b3M6fkc=NiMvIiInKiYtJSFHNiMiIiMiIiItJSFHNiMiIiQiIiI=NiMvIiNOKiYtJSFHNiMiIiYiIiItJSFHNiMiIigiIiI=NiMvIiRAIyomLSUhRzYjIiM4IiIiLSUhRzYjIiM8IiIi

Assim como os estudos todos os n\303\272meros naturais de grande magnitude s\303\243o primos ou compostos. Exemplificaremos com um primeiro conjunto desses n\303\272meros inteiros - os n\303\272meros de 2 a 40 Inteiros := { seq(k, k = 2..40) } ;

NiM+JSlJbnRlaXJvc0c8SSIiIyIiJCIiJSIiJiIiJyIiKCIiKSIiKiIjNSIjNiIjNyIjOCIjOSIjOiIjOyIjPCIjPSIjPiIjPyIjQCIjQSIjQiIjQyIjRCIjRSIjRiIjRyIjSCIjSSIjSiIjSyIjTCIjTSIjTiIjTyIjUCIjUSIjUiIjUw==

Esses s\303\243o os n\303\272meros primos \302\262 40. Perceba que nenhum deles pode ser fatorado. Primos := select( is Primos, Integers ); for k from 1 to nops(Primos) do Primos[k] = ifactor(Primos[k]); od;

Error, missing operator or `;`

Esses s\303\243o os n\303\272meros divis\303\255veis \302\262 40. Perceba que cada um deles pode ser fatorado. Composites := remove( is prime, Integers);

Error, missing operator or `;`

for k from 1 to nops(Composites) do Composites[k] = ifactor(Composites[k]); od;

Error, (in ifactor) invalid arguments

Existe um m\303\251todo antigo para se encontrar v\303\241rios n\303\272meros primos de uma vez. Ele come\303\247a com uma tabela de n\303\272meros naturais, e depois elimina os m\303\272ltiplos dos primos. Os primos em si devem permanecer mas, deve-se remover todos os m\303\272ltiplos daquele primo. Voc\303\252 s\303\263 precisa usar n\303\272meros menores ou iguais \303\240 raiz quadrada do maior n\303\272mero da sua tabela. Vamos procurar todos os primos at\303\251 40 ..... (Aten\303\247\303\243o : Ap\303\263s executar essa linha, voc\303\252 ter\303\241 que retornar manualmente para ver cada passo). Sieve(40);

NiMtJSZTaWV2ZUc2IyIjUw==

Neste diagrama, voc\303\252 tambem pode ver a distribui\303\247\303\243o de n\303\272merosprimos neste diagrama. Observe os "primos consecutivos" : (3,5), (5,7),(11,13), (17,19), (29,31),(41,43). Pergunta: Quantos s\303\243o os primos menores que 46? Veja o diagrama e conte quantos n\303\272meros tem. Voc\303\252 tamb\303\251m pode executar a proxima linha. numtheory[pi](46);

NiMiIzk=

Se voc\303\252 estiver pronto para um pouco mais de a\303\247\303\243o, tente o pr\303\263ximo comando ... e segure-se! Sieve(199);

NiMtJSZTaWV2ZUc2IyIkKj4=

Aqui voc\303\252 tamb\303\251m pode ver a distribui\303\247\303\243o dos primos menores que 200 ! Perceba que eles tendem a ficar mais espa\303\247ados \303\240 medida que os n\303\272meros aumentam - em geral. Entretanto, ainda vemos "Primos consecutivos" como 191 e 193 ou 197 e 199 mesmo pr\303\263ximos ao fim! Voc\303\252 sabe quantos n\303\272meros tem nessa tabela? Voc\303\252 deveria contar esse n\303\272mero: numtheory[pi](200);

NiMiI1k=

Tamb\303\251m podemos estudar a distribui\303\247\303\243o dos primos de outra maneira. Se contarmos quantos primos s\303\243o menores ou iguais que um n\303\272mero dado, e fizermos um gr\303\241fico dessa conta. Ent\303\243o, obteremos esse gr\303\241fico bem interessante: plot( numtheory[pi](floor(x)), x = 2..120);

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NiI=

Parece haver um padr\303\243o...que \303\251 a conjetura de Gauss. plot( {numtheory[pi](floor(x)), x/ln(x)}, x = 2..120);

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Assim como a F\303\255sica descobriu estruturas menores como os \303\241tomos e a Biologia como as c\303\251lulas, a Matem\303\241tica tamb\303\251m descobre n\303\272meros menores. Neste sentido, n\303\272meros divis\303\255veis s\303\243o aqueles compostos de n\303\272meros menores, enquanto n\303\272meros primos s\303\243o compostos apenas deles pr\303\263prios . Enquanto a F\303\255sica n\303\243o consegue ir al\303\251m dos quarks a Matem\303\241tica n\303\243o consegue ir al\303\251m dos n\303\272meros primos. Vamos estudar essa "Mat\303\251ria Matem\303\241tica" (conjunto): 2100 = `(30)(70)`;

NiMvIiUrQCUpKDMwKSg3MClH

Note que, cada um dos fatores \303\251 divis\303\255vel, j\303\241 que tamb\303\251m podem ser fatorados. 30 = `(6)(5)`; 70 = `(7)(10)`;

NiMvIiNJJScoNikoNSlHNiMvIiNxJSgoNykoMTApRw==

E al\303\251m disso, alguns desses fatores s\303\243o divis\303\255veis e ainda podem continuar serem fatorados. 6 = `(3)(2)`; 10 = `(5)(2)`;

NiMvIiInJScoMykoMilHNiMvIiM1JScoNSkoMilH

considerando que cada n\303\272mero divis\303\255vel pode ser fatorado, e possivelmente esses fatores continuarem a ser fatorados, esse processo deve ter um fim. O processo termina quando todos fatores de um n\303\272mero s\303\243o primos. Enquanto um fator n\303\243o for primo, ele ainda pode ser fatorado. 2100 = `(30)(70) = (6)(5)(7)(10) = (3)(2)(5)(7)(5)(2)`;

NiMvIiUrQCVOKDMwKSg3MCl+PX4oNikoNSkoNykoMTApfj1+KDMpKDIpKDUpKDcpKDUpKDIpRw==

Assim, ap\303\263s reunirmos todos os possiveis primos, e posicion\303\241-los em ordem crescente, teremos uma fatora\303\247\303\243o de primos para cada n\303\272mero divis\303\255vel: ifactor(2100);

NiMqKiktJSFHNiMiIiMiIiMiIiItJSFHNiMiIiQiIiIpLSUhRzYjIiImIiIjIiIiLSUhRzYjIiIoIiIi

Podemos facilmente obter a fatora\303\247\303\243o de primos em qualquer n\303\272mero divis\303\255vel usando o comando do Maple para fatora\303\247\303\243o de n\303\272meros inteiros, ifactor. 32: % = ifactor(%); 105: % = ifactor(%); 192: % = ifactor(%);

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432: % = ifactor(%); 4332: % = ifactor(%); 4320: % = ifactor(%);

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Na pr\303\241tica, voce pode utilizar uma "\303\241rvore" de fatores para fatorar um n\303\272mero completamente, e ent\303\243o, juntar todas as "folhas" da \303\241rvore que s\303\243o o mesmo n\303\272mero primo, para depois expressar na forma exponencial.

Parece que os n\303\272meros primos s\303\243o infinitos: Esses s\303\243o os primeiros 20 n\303\272meros primos .... Primes(20);

NiMtJSdQcmltZXNHNiMiIz8=

e, esses s\303\243o os primeiros 200 n\303\272meros primos ... Primes(100);

NiMtJSdQcmltZXNHNiMiJCsi

Esses s\303\243o os primeiros 250 n\303\272meros primos .... Primes(250);

NiMtJSdQcmltZXNHNiMiJF0j

e, tamb\303\251m podemos encontrar o cent\303\251simo n\303\272mero primo: ithprime(100);

NiMiJFQm

ou o ducent\303\251simo, tricent\303\251simo, quatrocent\303\251simo ..., e os n\303\272meros primos assim por diante for k from 1 to 20 do print( cat(100*k, `\302\272 primo \303\251 `, ithprime(100*k))); od;

NiMlNTEwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+NTQxRw==NiMlNjIwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+MTIyM0c=NiMlNjMwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+MTk4N0c=NiMlNjQwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+Mjc0MUc=NiMlNjUwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+MzU3MUc=NiMlNjYwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+NDQwOUc=NiMlNjcwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+NTI3OUc=NiMlNjgwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+NjEzM0c=NiMlNjkwMHxqeXxqdnxodn5wcmltb358anl8anZ8aHZ+Njk5N0c=NiMlNzEwMDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjc5MTlHNiMlNzExMDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjg4MzFHNiMlNzEyMDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjk3MzNHNiMlODEzMDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjEwNjU3Rw==NiMlODE0MDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjExNjU3Rw==NiMlODE1MDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjEyNTUzRw==NiMlODE2MDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjEzNDk5Rw==NiMlODE3MDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjE0NTE5Rw==NiMlODE4MDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjE1NDAxRw==NiMlODE5MDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjE2MzgxRw==NiMlODIwMDB8anl8anZ8aHZ+cHJpbW9+fGp5fGp2fGh2fjE3Mzg5Rw==

Pergunta que n\303\243o ofende:existe um n\303\272mero primo ainda maior? Digamos que exista. Digamos que p \303\251 o maior n\303\272mero primo. E ent\303\243o se multiplicarmos todos os n\303\272meros primos juntos, e somarmos um: N = 1 + p1*p2*p3*.....*pn Esse novo n\303\272mero nao \303\251 divis\303\255vel como qualquer n\303\272mero primo - o que \303\251 contradit\303\263rio. Logo devem existir mais n\303\272meros primos do que o suposto maior n\303\272mero primo. Isso significa que existem mesmo um n\303\272mero infinito de primos. Vamos tentar ver isso com alguns primos. Consideramos que 7 fosse o maior primo. LargestPrime(7);

NiMtJS1MYXJnZXN0UHJpbWVHNiMiIig=

e se pensasemos que 17 fosse o maior n\303\272mero primo? ou 37, ou 53? (Aten\303\247\303\243o : quanto maior o n\303\272mero, mais tempo vai demorar. Se demorar muito, tecle o sinal de Pare na barra de ferramentas no topo da tela do Maple.) LargestPrime(17);

NiMtJS1MYXJnZXN0UHJpbWVHNiMiIzw=

LargestPrime(37);

NiMtJS1MYXJnZXN0UHJpbWVHNiMiI1A=

LargestPrime(53);

NiMtJS1MYXJnZXN0UHJpbWVHNiMiI2A=

Portanto h\303\241 um n\303\272mero infinito de n\303\272meros primos, pois por maior que seja o n\303\272mero que encontremos, sempre haver\303\241 um maior ! Os matem\303\241ticos j\303\241 encontraram n\303\272meros primos com centenas de milhares de d\303\255gitos. Quantas p\303\241ginas seriam necess\303\241rias para escrever um n\303\272mero de 300,000 d\303\255gitos? Fa\303\247amos uma experi\303\252ncia. Usando o meu computador (um PC ), eu acho que esse n\303\272mero \303\251 a maior pot\303\252ncia de dez que eu posso digitar no meu computador em uma s\303\263 linha. Digamos que tenhamos 52 linhas por p\303\241gina. 1*10^105;

NiMiZXErKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrNQ==

`n\303\272meros de digitos` = 300000; `n\303\272meros de linhas` = %/105; `n\303\272meros de p\303\241ginas` = %/50;

NiMvJTVufGp5fGp2fGh2bWVyb3N+ZGV+ZGlnaXRvc0ciJysrSQ==NiMvJTRufGp5fGp2fGh2bWVyb3N+ZGV+bGluaGFzRy8sJComIiQwIiEiIiU1bnxqeXxqdnxodm1lcm9zfmRlfmRpZ2l0b3NHIiIiIiIiIyImKysjIiIoNiMvJTdufGp5fGp2fGh2bWVyb3N+ZGV+cHxqeXxqdnxodmdpbmFzRy8sJComIiNdISIiJTRufGp5fGp2fGh2bWVyb3N+ZGV+bGluaGFzRyIiIiIiIi8sJComIiVdXyEiIiU1bnxqeXxqdnxodm1lcm9zfmRlfmRpZ2l0b3NHIiIiIiIiIyIkKyUiIig=

evalf((300000)/(105*52));

NiMkIislXDBYXCYhIik=

Seriam necess\303\241rias 55 p\303\241ginas apenas para escrever um desses grandes n\303\272meros primos. Acho melhor n\303\243o tentarmos encontrar e visualizar esses n\303\272meros, mas simplesmente viver sabendo que esses n\303\272meros realmente existem.

Simpleficarmos ra\303\255zes quadradas, deixaremos dentro do radical apenas os n\303\272meros que n\303\243o s\303\243o quadrados perfeitos. . Esses n\303\272meros n\303\243o apresentam ra\303\255zes quadradas exatas. Observe os seguintes n\303\272meros por um instante. 42 : % = ifactor(%); 330 : % = ifactor(%); 2431 : % = ifactor(%);

NiMvIiNVKigtJSFHNiMiIiMiIiItJSFHNiMiIiQiIiItJSFHNiMiIigiIiI=NiMvIiRJJCoqLSUhRzYjIiIjIiIiLSUhRzYjIiIkIiIiLSUhRzYjIiImIiIiLSUhRzYjIiM2IiIiNiMvIiVKQyooLSUhRzYjIiM2IiIiLSUhRzYjIiM4IiIiLSUhRzYjIiM8IiIi

FindSquare(42); FindSquare(330); FindSquare(2431);

NiMlOTQyfm58anl8anZ8aHZvfnRlbX5yYWl6fmV4YXRhIUc=NiMlOjMzMH5ufGp5fGp2fGh2b350ZW1+cmFpen5leGF0YSFHNiMlOzI0MzF+bnxqeXxqdnxodm9+dGVtfnJhaXp+ZXhhdGEhRw==

Cada um desses n\303\272meros \303\251 fator\303\241vel - mas todos os fatores t\303\252m o expoente 1. Obviamente nenhum primo \303\251 quadrado perfeito. 17 : % = ifactor(%); 41 : % = ifactor(%); 97 : % = ifactor(%);

NiMvIiM8LSUhRzYjIiM8NiMvIiNULSUhRzYjIiNUNiMvIiMoKi0lIUc2IyIjKCo=

FindSquare(17); FindSquare(41); FindSquare(97);

NiMlOTE3fm58anl8anZ8aHZvfnRlbX5yYWl6fmV4YXRhIUc=NiMlOTQxfm58anl8anZ8aHZvfnRlbX5yYWl6fmV4YXRhIUc=NiMlOTk3fm58anl8anZ8aHZvfnRlbX5yYWl6fmV4YXRhIUc=

Agora, vamos observar esses n\303\272meros, que n\303\243o s\303\243o quadrados perfeitos, por\303\251m, s\303\243o fator\303\241veis. Voc\303\252 consegue identificar quadrados perfeitos "escondidos" entre os fatores? 12 : % = ifactor(%); 375 : % = ifactor(%); 72 : % = ifactor(%); 686 : % = ifactor(%);

NiMvIiM3KiYpLSUhRzYjIiIjIiIjIiIiLSUhRzYjIiIkIiIiNiMvIiR2JComLSUhRzYjIiIkIiIiKS0lIUc2IyIiJiIiJCIiIg==NiMvIiNzKiYpLSUhRzYjIiIjIiIkIiIiKS0lIUc2IyIiJCIiIyIiIg==NiMvIiQnbyomLSUhRzYjIiIjIiIiKS0lIUc2IyIiKCIiJCIiIg==

FindSquare(12); FindSquare(375); FindSquare(72); FindSquare(686);

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Observe que os n\303\272meros sem raiz quadrada s\303\243o sempre produtos de n\303\272meros primos distintos. Se um n\303\272mero primo tiver um quadrado, um cubo, a quarta pot\303\252ncia, etc ..... ent\303\243o ele cont\303\251m no m\303\255nimo um quadrado perfeito. 6 : % = ifactor(%); FindSquare(%%);

NiMvIiInKiYtJSFHNiMiIiMiIiItJSFHNiMiIiQiIiI=NiMlODZ+bnxqeXxqdnxodm9+dGVtfnJhaXp+ZXhhdGEhRw==

12 : % = ifactor(%); FindSquare(%%);

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24 : % = ifactor(%); FindSquare(%%);

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48 : % = ifactor(%); FindSquare(%%);

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96: % = ifactor(%); FindSquare(%%);

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192 : % = ifactor(%); FindSquare(%%);

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