Pengguna:Klasüo/bak pasir/khusus/1

Bilangan komposit yang disusun menjadi persegi panjang sedangkan bilangan prima tidak

Sebuah bilangan prima (atau prima) adalah bilangan besar dari 1 yang bukan bagian produk dari dua bilangan kecil. Bilangan besar dari 1 bukanlah prima yang disebut bilangan gabungan. Misalnya, 5 adalah prima karena satu-satunya cara untuk menulisnya sebagai produk, $1\times 5$ atau $5\times 1$, melibatkan 5 itu sendiri. Namun, 4 juga termasuk komposit karena merupakan produk ($2\times 2$) yang dimana kedua bilangan tersebut adalah kecil dari 4. Bilangan prima adalah pusat dalam teori bilangan karena termasuk dalam teorema dasar aritmetika: setiap bilangan besar dari 1 adalah bilangan prima itu sendiri atau melakukan faktorisasi sebagai perkalian bilangan prima unik hingga pada urutannya.

Sifat yang menjadikan sebuah prima disebut primalitas. Metode sederhana namun lambat untuk memeriksa primalitas dari bilangan tertentu yaitu $n$, disebut pembagian percobaan, menguji apakah $n$ adalah kelipatan dari sembarang bilangan bulat antara 2 dan ${\sqrt {n}}$. Algoritma yang mencakup uji primalitas Miller–Rabin, yang cepat tetapi memiliki peluang kesalahan yang kecil, dan uji primalitas AKS, yang selalu menghasilkan jawaban yang benar dalam waktu polinomial tetapi sangat lambat untuk praktis. Metode yang cepat tersedia untuk sejumlah bentuk khusus, seperti bilangan Mersenne. Hingga Desember 2018 Bilangan prima terbesar diketahui adalah bilangan prima Mersenne dengan 24.862.048 digit desimal.

Terdapat bilangan prima secara tak hingga, seperti yang ditunjukkan oleh Euklides sekitar 300 SM. Tidak ada rumus sederhana yang diketahui memisahkan bilangan prima dari bilangan komposit. Namun, distribusi bilangan prima dalam bilangan asli dalam jumlah besar dimodelkan secara statistik. Hasil pertama dari arah tersebut adalah teorema bilangan prima yang telah dibuktikan pada akhir abad ke-19, yang menyatakan bahwa probabilitas dari bilangan prima dipilih secara acak adalah proporsional berbanding balik dengan jumlah digitnya, yaitu logaritma.

Beberapa pertanyaan sejarah tentang bilangan prima masih belum terpecahkan. Hal ini termasuk dalam konjektur Goldbach, bahwa setiap bilangan bulat genap yang lebih besar dari 2 dapat dinyatakan sebagai jumlah dari dua bilangan prima, dan konjektur prima kembar, bahwa terdapat lebih pasangan bilangan prima yang memiliki satu bilangan genap diantara mereka. Pertanyaan-pertanyaan seperti itu mendorong pengembangan berbagai cabang teori bilangan, dengan fokus pada analitik atau aljabar aspek bilangan. Bilangan prima digunakan dalam beberapa rutinitas dalam teknologi informasi, seperti kriptografi kunci publik, yang bergantung pada kesulitan faktor bilangan besar menjadi faktor primanya. Dalam aljabar abstrak, objek yang berjalan secara umum seperti bilangan prima termasuk elemen prima dan ranah prima.

Definisi dan contoh-contohnya

Artikel utama: Daftar bilangan prima

Bilangan asli (1, 2, 3, 4, 5, 6, dll.) disebut juga bilangan prima (atau prima) jika bilangan tersebut lebih besar dari 1 dan tidak ditulis sebagai produk dari dua bilangan asli yang lebih kecil. Bilangan yang lebih besar dari 1 bukanlah prima yang disebut bilangan gabungan. Dengan kata lain, $n$ adalah prima jika $n$ yang tidak dapat dibagi menjadi kelompok-kelompok yang lebih kecil dengan ukuran yang sama dari satu item, atau jika tidak memungkinkan untuk menyusun $n$ pada titik menjadi kotak persegi panjang yang lebarnya lebih dari satu titik dan tingginya lebih dari satu titik. Misalnya, antara bilangan 1 sampai 6, bilangan 2, 3, dan 5 adalah bilangan prima, karena tidak ada bilangan lain yang membagi secara merata (tanpa sisa). 1 bukan prima, karena secara khusus dikecualikan dalam definisi. $4=2\times 2$ dan $6=2\times 3$ keduanya adalah komposit.

Demonstrasi, dengan batang Cuisenaire, bahwa 7 adalah prima, karena tidak ada 2, 3, 4, 5, atau 6 yang membaginya secara merata

Pembagi dari bilangan asli $n$ adalah bilangan asli yang membagi $n$ secara merata. Setiap bilangan asli memiliki 1 dan dirinya sendiri sebagai pembagi. Jika memiliki pembagi lain, maka itu tidak bisa sebagai prima. Ide ini mengarah ke definisi yang berbeda tetapi setara dari bilangan prima: mereka adalah bilangan dengan tepat dua pembagi positif, 1, dan bilangan itu sendiri. Cara lain untuk menyatakan hal yang sama adalah bahwa suatu bilangan $n$ adalah prima jika lebih besar dari satu dan jika tidak satupun dari bilangan $2,3,\dots ,n-1$ membagi $n$ secara merata.

25 bilangan prima pertama (semua bilangan prima kurang dari 100) adalah:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (barisan A000040 pada OEIS).

Tidak, bilangan genap $n$ yang lebih besar dari 2 adalah prima karena bilangan sembarang dapat dinyatakan sebagai hasil kali $2\times n/2$. Oleh karena itu, setiap bilangan prima selain 2 adalah bilangan ganjil, dan disebut sebagai prima ganjil. Demikian pula, ketika ditulis dalam sistem desimal biasa, semua bilangan prima yang lebih besar dari 5 diakhiri dengan 1, 3, 7, atau 9. Angka-angka yang diakhiri dengan digit lain semuanya komposit: bilangan desimal dengan hasil 0, 2, 4, 6, atau 8 adalah genap, dan bilangan desimal dengan hasil 0 atau 5 habis dibagi 5.

Himpunan dari semua bilangan prima terkadang dilambangkan dengan $\mathbf {P}$ (dengan huruf tebal kapital P) atau dengan $\mathbb {P}$ (dengan papan tulis tebal kapital P).

Sejarah

Papirus Matematika Rhind

Papirus Matematika Rhind dari sekitar tahun 1550 SM, memiliki pecahan Mesir ekspansi bentuk yang berbeda untuk bilangan prima dan komposit. Namun, catatan pertama kali yang bertahan studi eksplisit bilangan prima berasal dari matematika Yunani kuno. Elemen dari Euklides (300 SM) membuktikan bilangan prima tak-hingga dan teorema dasar aritmetika, dan menunjukkan cara membuat bilangan sempurna dari prima Mersenne. Penemuan Yunani lainnya yaitu tapis Eratosthenes masih digunakan untuk menyusun daftar bilangan prima.

Sekitar 1000 M, matematikawan Islam Ibn al-Haytham (Alhazen) menemukan teorema Wilson dengan mencirikan bilangan prima sebagai bilangan $n$ yang membagi rata $(n-1)!+1$. Ia juga menduga bahwa semua bilangan sempurna genap berasal dari konstruksi Euklides yang menggunakan bilangan prima Mersenne, tetapi tidak dapat membuktikannya. Matematikawan Islam lainnya, Ibn al-Banna' al-Marrakushi mengamati bahwa pitas Eratosthenes dapat dipercepat dengan menguji hanya pembagi hingga akar kuadrat dari bilangan terbesar yang akan diuji. Fibonacci membawa inovasi dari matematika Islam kembali ke Eropa. Liber Abaci (1202) dalam bukunya yang pertama mendeskripsikan pembagian percobaan untuk menguji primalitas, sekali lagi menggunakan pembagi hanya akar kuadrat hingga.

...

In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also investigated the primality of the Fermat numbers $2^{2^{n}}+1$, and Marin Mersenne studied the Mersenne primes, prime numbers of the form $2^{p}-1$ with $p$ itself a prime. Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes ${\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots$. At the start of the 19th century, Legendre and Gauss conjectured that as $x$ tends to infinity, the number of primes up to $x$ is asymptotic to $x/\log x$, where $\log x$ is the natural logarithm of $x$. A weaker consequence of this high density of primes was Bertrand's postulate, that for every $n>1$ there is a prime between $n$ and $2n$, proved in 1852 by Pafnuty Chebyshev. Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem. Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes.

...

Many mathematicians have worked on primality tests for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877), Proth's theorem (c. 1878), the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test.

Since 1951 all the largest known primes have been found using these tests on computers. The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects. The idea that prime numbers had few applications outside of pure mathematics was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.

The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form. The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size.

Primality of one

Most early Greeks did not even consider 1 to be a number, so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number. By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and some of them included it as the first prime number. In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime. In the 19th century many mathematicians still considered 1 to be prime, and lists of primes that included 1 continued to be published as recently as 1956.

If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1. Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1. By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".

Sifat dasar

Faktorisasi unik

Artikel utama: Teorema dasar aritmetika

Menulis bilangan sebagai produk bilangan prima disebut faktorisasi prima dari bilangan tersebut. Misalnya:

${\begin{aligned}34866&=2\cdot 3\cdot 3\cdot 13\cdot 149\\&=2\cdot 3^{2}\cdot 13\cdot 149.\end{aligned}}$

istilah dalam produk tersebut disebut juga sebagai faktor prima. Faktor prima yang sama dapat muncul lebih dari sekali; contoh ini memiliki dua salinan faktor prima $3.$ Ketika bilangan prima muncul beberapa kali, eksponensial digunakan untuk pengelompokkan beberapa salinan dari bilangan prima yang sama: misalnya, dalam cara kedua penulisan produk atas, $3^{2}$ menunjukkan persegi atau pangkat kedua dari $3.$

Pentingnya pusat bilangan prima untuk teori bilangan dan matematika secara umum berasal dari teorema dasar aritmetika. Teorema ini menyatakan bahwa setiap bilangan bulat yang lebih besar dari 1 ditulis sebagai produk dari satu atau lebih bilangan prima. Secara sederhana, produk ini adalah unik dalam arti bahwa dua faktorisasi prima dari bilangan yang sama akan memiliki jumlah salinan yang sama dari bilangan prima yang sama, meskipun urutannya mungkin berbeda. Jadi, meskipun terdapat berbagai banyak cara yang berbeda untuk menemukan faktorisasi menggunakan algoritma faktorisasi bilangan bulat, semuanya adalah hasil yang sama. Dengan demikian bilangan prima menganggap sebagai "blok bangunan dasar" dari bilangan asli.

Beberapa bukti keunikan faktorisasi prima didasarkan pada lemma Euklides: Jika $p$ adalah bilangan prima dan $p$ membagi hasil kali $ab$ dari bilangan bulat $a$ dan $b,$ maka $p$ membagi $a$ atau $p$ membagi $b$ (atau keduanya). Sebaliknya, jika suatu bilangan $p$ memiliki sifat bahwa ketika membagi produk selalu membagi setidaknya satu faktor dari produk, maka $p$ adalah prima.

Jumlah tak hingga

Artikel utama: Teorema Euklides

Cara lain untuk mengatakan urutannya adalah

2, 3, 5, 7, 11, 13, ...

bilangan prima tak-akhir. Pernyataan ini disebut sebagai teorema Euklides untuk menghormati matematikawan Yunani kuno Euklides, sejak bukti pertama yang diketahui untuk pernyataan yang dikaitkan dengan dia. Banyak lagi bukti bilangan prima tak hingga yang diketahui, termasuk bukti analitis oleh Euler, bukti Goldbach berdasarkan bilangan Fermat, bukti menggunakan topologi umum Furstenberg, and Kummer's elegant proof.

Bukti Euclid menunjukkan bahwa setiap daftar hingga dari bilangan prima tak-lengkap. Ide kuncinya adalah mengalikan bilangan prima dalam daftar yang diberikan dan menambahkan $1.$ Jika daftar tersebut terdiri dari bilangan prima $p_{1},p_{2},\ldots ,p_{n},$ maka diberikan bilangan

$N=1+p_{1}\cdot p_{2}\cdots p_{n}.$

Berdasarkan teorema dasar, $N$ memiliki faktorisasi prima

$N=p'_{1}\cdot p'_{2}\cdots p'_{m}$

dengan satu atau lebih dari faktor prima. Maka, $N$ habis dibagi nilai rata-rata oleh faktor ini, tetapi $N$ memiliki sisa satu ketika dibagi dengan salah satu bilangan prima dalam daftar yang diberikan, jadi tidak ada faktor prima $N$ yang ada pada daftar yang diberikan. Karena tidak ada daftar hingga dari semua bilangan prima, pasti ada banyak bilangan prima.

Bilangan yang dibentuk dengan menjumlahkan satu ke hasil kali bilangan prima terkecil disebut bilangan Euklides. Lima bagian pertama adalah bilangan prima, namun untuk bagian keenam

$1+{\big (}2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13{\big )}=30031=59\cdot 509,$

adalah bilangan komposit.

Rumus bilangan prima

Artikel utama: Rumus bilangan prima

Tidak ada rumus efisien yang diketahui untuk bilangan prima. Misalnya, tidak ada polinomial non-konstan, bahkan dalam beberapa variabel mengambil hanya bilangan prima. Namun, ada banyak ekspresi yang mengkodekan semua bilangan prima atau hanya bilangan prima. Satu rumus yang mungkin didasarkan pada teorema Wilson dan menghasilkan angka 2 berkali-kali dan semua bilangan prima lainnya tepat. Ada pula satu himpunan persamaan Diophantine dalam sembilan variabel dan satu parameter dengan sifat berikut: parameter bilangan prima jika dan hanya jika sistem persamaan yang dihasilkan memiliki solusi atas bilangan asli. Hal ini digunakan untuk mendapatkan rumus tunggal dengan sifat bahwa semua nilai "positif" adalah prima.

Contoh lain dari rumus pembangkit-prima berasal dari teorema Mills dan teorema Wright. Maka ini menegaskan bahwa terdapat konstanta real $A>1$ dan $\mu$ sedemikian rupa, sehingga

$\left\lfloor A^{3^{n}}\right\rfloor {\text{ dan }}\left\lfloor 2^{\cdots ^{2^{2^{\mu }}}}\right\rfloor$

adalah prima untuk sembarang bilangan asli $n$ dalam rumus pertama, dan sembarang bilangan eksponen dalam rumus kedua. Sehingga $\lfloor {}\cdot {}\rfloor$ mewakili fungsi lantai, bilangan bulat terbesar yang kurang dari atau sama dengan bilangan yang dimaksud. Namun, hal ini justru tidak berguna untuk menghasilkan bilangan prima, karena bilangan prima harus dibangkitkan terlebih dahulu untuk menghitung nilai $A$ atau $\mu .$

Pertanyaan terbuka

Informasi lebih lanjut: Kategori:Konjektur tentang bilangan prima

Banyak konjektur tentang bilangan prima telah diajukan. Seringkali memiliki rumus dasar, banyak dari konjektur ini telah bertahan selama beberapa dekade: keempat masalah Landau dari tahun 1912 masih belum terpecahkan. Salah satunya adalah konjektur Goldbach yang menyatakan bahwa setiap bilangan bulat genap $n$ lebih besar dari 2 ditulis sebagai jumlah dari dua bilangan prima. Hingga 2014, Konjektur ini telah diverifikasi untuk semua bilangan hingga $n=4\cdot 10^{18}.$ Pernyataan yang lebih lemah dari ini telah dibuktikan, misalnya teorema Vinogradov yang menyatakan bahwa setiap bilangan bulat ganjil besar dapat ditulis sebagai jumlah dari tiga bilangan prima. Teorema Chen menyatakan bahwa setiap bilangan genap besar dapat dinyatakan sebagai jumlah dari suatu bilangan prima dan semiprima (perkalian dari dua bilangan prima). Juga, bilangan bulat genap lebih besar dari 10 dapat ditulis sebagai jumlah dari enam bilangan prima. Cabang teori bilangan yang mempelajari pertanyaan semacam itu disebut teori bilangan aditif.

Jenis masalah lain menyangkut celah prima, perbedaan antara bilangan prima berurutan. Adanya celah prima besar secara sembarang dapat dilihat dengan memperhatikan bahwa barisan $n!+2,n!+3,\dots ,n!+n$ terdiri dari $n-1$ bilangan komposit, untuk sembarang bilangan asli $n.$ However, large prime gaps occur much earlier than this argument shows. Misalnya, celah prima pertama dengan panjang 8 adalah antara bilangan prima 89 dan 97, jauh lebih kecil dari $8!=40320.$ Diduga ada banyak sekali prima kembars, pasangan bilangan prima dengan selisih 2; ini adalah konjektur prima kembar. Konjektur Polignac menyatakan secara lebih umum bahwa untuk setiap bilangan bulat positif $k,$ ada tak hingga banyak pasangan bilangan prima berurutan yang berbeda $2k.$ Konjektur Andrica, Brocard's conjecture, Legendre's conjecture, and Oppermann's conjecture semua menyarankan bahwa jarak terbesar antara bilangan prima dari $1$ hingga $n$ paling banyak kira-kira ${\sqrt {n}},$ hasil yang diketahui mengikuti hipotesis Riemann, sedangkan konjektur Cramér lebih bertahan menetapkan ukuran celah terbesar pada $O((\log n)^{2}).$ Celah prima digeneralisasikan ke rangkap-$k$ prima, pola selisih antara lebih dari dua bilangan prima. Ketakhinggaan dan kepadatan mereka adalah subjek dari konjektur Hardy–Littlewood, yang dapat dimotivasi oleh heuristik bahwa bilangan prima berperilaku serupa dengan barisan bilangan acak dengan kerapatan yang diberikan oleh teorema bilangan prima.

Sifat analitik

Teori bilangan analitik mempelajari teori bilangan melalui lensa fungsi kontinu, limit, deret tak hingga, dan matematika terkait dari tak hingga dan infinitesimal.

Bidang studi ini dimulai dengan Leonhard Euler dan hasil besar pertamanya, solusi untuk masalah Basel. Soal menanyakan nilai jumlah tak hingga $1+{\tfrac {1}{4}}+{\tfrac {1}{9}}+{\tfrac {1}{16}}+\dots ,$ yang biasanya dikenali sebagai nilai $\zeta (2)$ dari fungsi Riemann zeta. Fungsi ini terkait erat dengan bilangan prima dan salah satu masalah paling signifikan yang belum terpecahkan dalam matematika, hipotesis Riemann. Euler menunjukkan bahwa $\zeta (2)=\pi ^{2}/6$. Kebalikan dari bilangan $6/\pi ^{2}$ adalah peluang limit bahwa dua bilangan acak yang dipilih secara seragam dari rentang yang besar adalah relatif prima (tidak memiliki faktor).

Distribusi bilangan prima dalam besar, seperti pertanyaan berapa banyak bilangan prima lebih kecil dari ambang besar yang diberikan, dijelaskan oleh teorema bilangan prima, tetapi tidak ada rumus untuk bilangan prima ke-$n$ yang memiliki efisien yang diketahui. Teorema Dirichlet pada barisan aritmetika dalam bentuk dasarnya menyatakan bahwa polinomial linear

$p(n)=a+bn$

dengan bilangan bulat relatif prima $a$ dan $b$ mengambil banyak nilai prima tak-hingga. Bentuk teorema lebih bertahan menyatakan bahwa jumlah kebalikan dari nilai-nilai prima ini adalah pembagian, dan polinomial linear berbeda dengan $b$ yang sama memiliki proporsi bilangan prima yang kira-kira sama. Meskipun konjektur telah dirumuskan tentang proporsi bilangan prima dalam polinomial tingkat tinggi mereka tetap tidak terbukti, dan tidak diketahui apakah polinomial kuadrat (untuk argumen bilangan bulat) adalah bilangan prima tak-hingga.

Bukti analitik teorema Euklides

Bukti Euler bahwa ada banyak bilangan prima yang mempertimbangkan jumlah kebalikan dari bilangan prima,

${\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+\cdots +{\frac {1}{p}}.$

Euler menunjukkan bahwa untuk sembarang bilangan reall $x$, terdapat sebuah $p$ prima yang jumlahnya lebih besar dari $x$. Ini menunjukkan bahwa ada banyak bilangan prima karena jika ada banyak bilangan prima, jumlah tersebut akan mencapai nilai maksimumnya pada bilangan prima terbesar dari bertambah melewati setiap $x$. Tingkat pertumbuhan jumlah ini dijelaskan lebih tepat oleh teorema kedua Mertens. Sebagai perbandingan, jumlah

${\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}$

tidak tumbuh secara tak-hingga saat $n$ menjadi tak-hingga (lihat masalah Basel). Dalam pengertian ini, bilangan prima sering muncul dibandingkan kuadrat bilangan asli, meskipun kedua himpunan tak-hingga. Teorema Brun menyatakan bahwa jumlah kebalikan dari prima kembar,

$\left({{\frac {1}{3}}+{\frac {1}{5}}}\right)+\left({{\frac {1}{5}}+{\frac {1}{7}}}\right)+\left({{\frac {1}{11}}+{\frac {1}{13}}}\right)+\cdots ,$

yang merupakan hingga. Karena teorema Brun, tidak mungkin menggunakan metode Euler untuk menyelesaikan konjektur prima kembar, bahwa terdapat banyak bilangan prima kembar tak-hingga.

Jumlah bilangan prima bawah batas yang diberikan

Artikel utama: Teorema bilangan prima dan Fungsi pencacahan prima

Kesalahan relatif dari ${\tfrac {n}{\log n}}$ dan integral logaritmik $\operatorname {Li} (n)$ sebagai aproksimasi untuk fungsi pencacahan prima. Kedua kesalahan relatif berkurang menjadi nol saat $n$ tumbuh, tetapi konvergensi ke nol jauh lebih cepat untuk integral logaritmik.

Fungsi pencacahan prima $\pi (n)$ didefinisikan sebagai jumlah bilangan prima tidak lebih besar dari $n$. Misalnya, $\pi (11)=5$, karena ada lima bilangan prima yang kurang dari atau sama dengan 11. Metode yang seperti algoritma Meissel–Lehmer menghitung nilai eksak $\pi (n)$ lebih cepat dibandingkan untuk membuat daftar setiap bilangan prima hingga $n$. Teorema bilangan prima menyatakan bahwa $\pi (n)$ adalah asimtotik terhadap $n/\log n$, yang dinotasikan sebagai

$\pi (n)\sim {\frac {n}{\log n}},$

dan berarti rasio $\pi (n)$ terhadap pecahan kanan pendekatan 1 saat $n$ bertambah secara tak-hingga. Ini menyiratkan bahwa kemungkinan bahwa bilangan yang dipilih secara acak kurang dari $n$ adalah prima adalah (kurang-lebih) berbanding kebalikan dengan jumlah digit dalam $n$. Ini juga menyiratkan bahwa bilangan prima ke $n$ sebanding dengan $n\log n$ dan oleh karena itu ukuran rata-rata celah prima sebanding dengan $\log n$. Estimasi yang akurat untuk $\pi (n)$ diberikan oleh ofset logaritmik integral

$\pi (n)\sim \operatorname {Li} (n)=\int _{2}^{n}{\frac {dt}{\log t}}.$

Barisan aritmetika

Artikel utama: Teorema Dirichlet tentang barisan aritmatika dan Teorema Green–Tao

Sebuah barisan aritmetika adalah barisan bilangan hingga atau tak-hingga sehingga bilangan-bilangan berurutan dalam barisan tersebut semuanya memiliki selisih yang sama. Perbedaan ini disebut modulus dari barisan. Misalnya,

3, 12, 21, 30, 39, ...,

yang adalah barisan aritmetika tak-hingga dengan modulus 9. Dalam barisan aritmetika, semua bilangan memiliki sisa yang sama jika dibagi dengan modulus; contohnya, sisanya adalah 3. Karena modulus 9 dan sisanya 3 adalah kelipatan 3, demikian juga setiap elemen dalam barisan. Oleh karena itu, barisan ini hanya berisi satu bilangan prima, 3 itu sendiri. Secara umum, barisan tak-hingga-nya

$a,a+q,a+2q,a+3q,\dots$

dapat memiliki lebih dari satu bilangan prima hanya jika sisa $a$ dan modulus $q$ relatif prima. Jika mereka relatif prima, Teorema Dirichlet tentang barisan aritmatika menyatakan bahwa barisan tersebut memiliki banyak bilangan prima.

Bilangan prima dalam barisan aritmetika modulo 9. Setiap baris pita horizontal tipis menunjukkan salah satu dari sembilan kemungkinan barisan mod 9, dengan bilangan prima ditandai dengan warna merah. Barisan bilangan 0, 3, atau 6 mod 9 mengandung paling banyak satu bilangan prima (bilangan 3); sisa barisan bilangan 2, 4, 5, 7, dan 8 mod 9 memiliki banyak bilangan prima, dengan bilangan prima yang sama pada setiap barisannya

Teorema Green–Tao menunjukkan bahwa ada barisan aritmetika hingga yang panjangnya yang terdiri dari bilangan prima.

Nilai prima polinomial kuadrat

Spiral Ulam. Bilangan prima (merah) mengelompokkan pada beberapa diagonal dan tidak pada yang lain. Nilai prima dari $4n^{2}-2n+41$ ditampilkan dengan warna biru.

Euler mencatat bahwa fungsi

$n^{2}-n+41$

menghasilkan bilangan prima untuk $1\leq n\leq 40$, meskipun bilangan komposit muncul antara nilai-nilai selanjutnya. Pencarian penjelasan untuk fenomena ini mengarah pada teori bilangan aljabar yang mendalam dari bilangan Heegner dan masalah bilangan kelas. Konjektur F Hardy-Littlewood memprediksi kerapatan bilangan prima antara nilai-nilai polinomial kuadrat dengan bilangan bulat koefisien dalam hal integral logaritmik dan polinomial. Tidak ada polinomial kuadrat yang terbukti memiliki banyak nilai prima secara tak-hingga.

Spiral Ulam mengatur bilangan asli dalam kisi dua dimensi, berputar dalam kotak konsentris yang mengelilingi asal dengan bilangan prima disorot. Secara visual, bilangan prima tampak mengelompokkan pada diagonal tertentu dan bukan pada diagonal lainnya, menunjukkan bahwa beberapa polinomial kuadrat mengambil nilai prima lebih sering daripada yang lainnya.

Fungsi Zeta dan hipotesis Riemann

Artikel utama: Hipotesis Riemann

Plot nilai absolut dari fungsi zeta, menunjukkan beberapa fiturnya

Salah satu pertanyaan tak terpecahkan paling terkenal dalam matematika yang berasal dari tahun 1859, dan salah satunya dari Masalah Hadiah Milenium adalah Hipotesis Riemann, yang dimana nol dari fungsi Riemann zeta $\zeta (s)$ berada. Fungsi ini adalah fungsi analitik pada bilangan kompleks. Untuk bilangan kompleks $s$ dengan bagian real lebih besar dari satu sama dengan jumlah tak hingga pada semua bilangan bulat, dan darab tak hingga diatas bilangan prima,

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}.$

This equality between a sum and a product, discovered by Euler, is called an Euler product. The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers. It leads to another proof that there are infinitely many primes: if there were only finitely many, then the sum-product equality would also be valid at $s=1$, but the sum would diverge (it is the harmonic series $1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\dots$) while the product would be finite, a contradiction.

The Riemann hypothesis states that the zeros of the zeta-function are all either negative even numbers, or complex numbers with real part equal to 1/2. The original proof of the prime number theorem was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1, although other more elementary proofs have been found. The prime-counting function can be expressed by Riemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term. In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of $x$ for intervals near a number $x$).

Abstract algebra

Modular arithmetic and finite fields

Artikel utama: Modular arithmetic

Modular arithmetic modifies usual arithmetic by only using the numbers $\{0,1,2,\dots ,n-1\}$, for a natural number $n$ called the modulus. Any other natural number can be mapped into this system by replacing it by its remainder after division by $n$. Modular sums, differences and products are calculated by performing the same replacement by the remainder on the result of the usual sum, difference, or product of integers. Equality of integers corresponds to congruence in modular arithmetic: $x$ and $y$ are congruent (written $x\equiv y$ mod $n$) when they have the same remainder after division by $n$. However, in this system of numbers, division by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number $7$ as modulus, division by $3$ is possible: $2/3\equiv 3{\bmod {7}}$, because clearing denominators by multiplying both sides by $3$ gives the valid formula $2\equiv 9{\bmod {7}}$. However, with the composite modulus $6$, division by $3$ is impossible. There is no valid solution to $2/3\equiv x{\bmod {6}}$: clearing denominators by multiplying by $3$ causes the left-hand side to become $2$ while the right-hand side becomes either $0$ or $3$. In the terminology of abstract algebra, the ability to perform division means that modular arithmetic modulo a prime number forms a field or, more specifically, a finite field, while other moduli only give a ring but not a field.

Several theorems about primes can be formulated using modular arithmetic. For instance, Fermat's little theorem states that if $a\not \equiv 0$ (mod $p$), then $a^{p-1}\equiv 1$ (mod $p$). Summing this over all choices of $a$ gives the equation

$\sum _{a=1}^{p-1}a^{p-1}\equiv (p-1)\cdot 1\equiv -1{\pmod {p}},$

valid whenever $p$ is prime. Giuga's conjecture says that this equation is also a sufficient condition for $p$ to be prime. Wilson's theorem says that an integer $p>1$ is prime if and only if the factorial $(p-1)!$ is congruent to $-1$ mod $p$. For a composite number $\;n=r\cdot s\;$ this cannot hold, since one of its factors divides both n and $(n-1)!$, and so $(n-1)!\equiv -1{\pmod {n}}$ is impossible.

p-adic numbers

Artikel utama: p-adic number

The $p$-adic order $\nu _{p}(n)$ of an integer $n$ is the number of copies of $p$ in the prime factorization of $n$. The same concept can be extended from integers to rational numbers by defining the $p$-adic order of a fraction $m/n$ to be $\nu _{p}(m)-\nu _{p}(n)$. The $p$-adic absolute value $|q|_{p}$ of any rational number $q$ is then defined as $|q|_{p}=p^{-\nu _{p}(q)}$. Multiplying an integer by its $p$-adic absolute value cancels out the factors of $p$ in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their $p$-adic distance, the $p$-adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of $p$. In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a complete field, the rational numbers with the $p$-adic distance can be extended to a different complete field, the $p$-adic numbers.

This picture of an order, absolute value, and complete field derived from them can be generalized to algebraic number fields and their valuations (certain mappings from the multiplicative group of the field to a totally ordered additive group, also called orders), absolute values (certain multiplicative mappings from the field to the real numbers, also called norms), and places (extensions to complete fields in which the given field is a dense set, also called completions). The extension from the rational numbers to the real numbers, for instance, is a place in which the distance between numbers is the usual absolute value of their difference. The corresponding mapping to an additive group would be the logarithm of the absolute value, although this does not meet all the requirements of a valuation. According to Ostrowski's theorem, up to a natural notion of equivalence, the real numbers and $p$-adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers. The local-global principle allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory.

Prime elements in rings

Artikel utama: Prime element dan Irreducible element

The Gaussian primes with norm less than 500

A commutative ring is an algebraic structure where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, prime elements and irreducible elements. An element $p$ of a ring $R$ is called prime if it is nonzero, has no multiplicative inverse (that is, it is not a unit), and satisfies the following requirement: whenever $p$ divides the product $xy$ of two elements of $R$, it also divides at least one of $x$ or $y$. An element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements form the same set,

$\{\dots ,-11,-7,-5,-3,-2,2,3,5,7,11,\dots \}\,.$

In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for unique factorization domains.

The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the Gaussian integers $\mathbb {Z} [i]$, the ring of complex numbers of the form $a+bi$ where $i$ denotes the imaginary unit and $a$ and $b$ are arbitrary integers. Its prime elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes $1+i$ and $1-i$. Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not. This is a consequence of Fermat's theorem on sums of two squares, which states that an odd prime $p$ is expressible as the sum of two squares, $p=x^{2}+y^{2}$, and therefore factorizable as $p=(x+iy)(x-iy)$, exactly when $p$ is 1 mod 4.

Prime ideals

Artikel utama: Prime ideals

Not every ring is a unique factorization domain. For instance, in the ring of numbers $a+b{\sqrt {-5}}$ (for integers $a$ and $b$) the number $21$ has two factorizations $21=3\cdot 7=(1+2{\sqrt {-5}})(1-2{\sqrt {-5}})$, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of an ideal, a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ... The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.

The spectrum of a ring is a geometric space whose points are the prime ideals of the ring. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. These concepts can even assist with in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers. The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by Chebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.

Group theory

In the theory of finite groups the Sylow theorems imply that, if a power of a prime number $p^{n}$ divides the order of a group, then the group has a subgroup of order $p^{n}$. By Lagrange's theorem, any group of prime order is a cyclic group, and by Burnside's theorem any group whose order is divisible by only two primes is solvable.

Computational methods

The small gear in this piece of farm equipment has 13 teeth, a prime number, and the middle gear has 21, relatively prime to 13

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of mathematics other than the use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.

This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public-key cryptography algorithms. These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining whether a given number is prime. The most basic primality testing routine, trial division, is too slow to be useful for large numbers. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for numbers of special types. Most primality tests only tell whether their argument is prime or not. Routines that also provide a prime factor of composite arguments (or all of its prime factors) are called factorization algorithms. Prime numbers are also used in computing for checksums, hash tables, and pseudorandom number generators.

Trial division

Artikel utama: Trial division

The most basic method of checking the primality of a given integer $n$ is called trial division. This method divides $n$ by each integer from 2 up to the square root of $n$. Any such integer dividing $n$ evenly establishes $n$ as composite; otherwise it is prime. Integers larger than the square root do not need to be checked because, whenever $n=a\cdot b$, one of the two factors $a$ and $b$ is less than or equal to the square root of $n$. Another optimization is to check only primes as factors in this range. For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to ${\sqrt {37}}$, which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime.

Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers. However, trial division is still used, with a smaller limit than the square root on the divisor size, to quickly discover composite numbers with small factors, before using more complicated methods on the numbers that pass this filter.

Sieves

The sieve of Eratosthenes starts with all numbers unmarked (gray). It repeatedly finds the first unmarked number, marks it as prime (dark colors) and marks its square and all later multiples as composite (lighter colors). After marking the multiples of 2 (red), 3 (green), 5 (blue), and 7 (yellow), all primes up to the square root of the table size have been processed, and all remaining unmarked numbers (11, 13, etc.) are marked as primes (magenta).

Artikel utama: Sieve of Eratosthenes

Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed. The oldest method for generating a list of primes is called the sieve of Eratosthenes. The animation shows an optimized variant of this method. Another more asymptotically efficient sieving method for the same problem is the sieve of Atkin. In advanced mathematics, sieve theory applies similar methods to other problems.

Primality testing versus primality proving

Some of the fastest modern tests for whether an arbitrary given number $n$ is prime are probabilistic (or Monte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer. For instance the Solovay–Strassen primality test on a given number $p$ chooses a number $a$ randomly from $2$ through $p-2$ and uses modular exponentiation to check whether $a^{(p-1)/2}\pm 1$ is divisible by $p$. If so, it answers yes and otherwise it answers no. If $p$ really is prime, it will always answer yes, but if $p$ is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2. If this test is repeated $n$ times on the same number, the probability that a composite number could pass the test every time is at most $1/2^{n}$. Because this decreases exponentially with the number of tests, it provides high confidence (although not certainty) that a number that passes the repeated test is prime. On the other hand, if the test ever fails, then the number is certainly composite. A composite number that passes such a test is called a pseudoprime.

In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. For instance, this is true of trial division. The algorithms with guaranteed-correct output include both deterministic (non-random) algorithms, such as the AKS primality test, and randomized Las Vegas algorithms where the random choices made by the algorithm do not affect its final answer, such as some variations of elliptic curve primality proving. When the elliptic curve method concludes that a number is prime, it provides primality certificate that can be verified quickly. The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based on heuristic arguments rather than rigorous proofs. The AKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice. These methods can be used to generate large random prime numbers, by generating and testing random numbers until finding one that is prime; when doing this, a faster probabilistic test can quickly eliminate most composite numbers before a guaranteed-correct algorithm is used to verify that the remaining numbers are prime.

The following table lists some of these tests. Their running time is given in terms of $n$, the number to be tested and, for probabilistic algorithms, the number $k$ of tests performed. Moreover, $\varepsilon$ is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that each time bound should be multiplied by a constant factor to convert it from dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters $n$ and $k$.

Test	Developed in	Type	Running time	Notes	References
AKS primality test	2002	deterministic	$O((\log n)^{6+\varepsilon })$
Elliptic curve primality proving	1986	Las Vegas	$O((\log n)^{4+\varepsilon })$ heuristically
Baillie–PSW primality test	1980	Monte Carlo	$O((\log n)^{2+\varepsilon })$
Miller–Rabin primality test	1980	Monte Carlo	$O(k(\log n)^{2+\varepsilon })$	error probability $4^{-k}$
Solovay–Strassen primality test	1977	Monte Carlo	$O(k(\log n)^{2+\varepsilon })$	error probability $2^{-k}$

Special-purpose algorithms and the largest known prime

Informasi lebih lanjut: List of prime numbers

In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. For example, the Lucas–Lehmer primality test can determine whether a Mersenne number (one less than a power of two) is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test. This is why since 1992 (hingga Desember 2018) the largest known prime has always been a Mersenne prime. It is conjectured that there are infinitely many Mersenne primes.

The following table gives the largest known primes of various types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.

Type	Prime	Number of decimal digits	Date	Found by
Mersenne prime	282,589,933 − 1	24,862,048	December 7, 2018	Patrick Laroche, Great Internet Mersenne Prime Search
Proth prime	10,223 × 231,172,165 + 1	9,383,761	October 31, 2016	Péter Szabolcs, PrimeGrid
factorial prime	208,003! − 1	1,015,843	July 2016	Sou Fukui
primorial prime	1,098,133# − 1	476,311	March 2012	James P. Burt, PrimeGrid
twin primes	2,996,863,034,895  × 21,290,000 ± 1	388,342	September 2016	Tom Greer, PrimeGrid

Integer factorization

Artikel utama: Integer factorization

Given a composite integer $n$, the task of providing one (or all) prime factors is referred to as factorization of $n$. It is significantly more difficult than primality testing, and although many factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division and Pollard's rho algorithm can be used to find very small factors of $n$, and elliptic curve factorization can be effective when $n$ has factors of moderate size. Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the quadratic sieve and general number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve. Hingga Desember 2019 the largest number known to have been factored by a general-purpose algorithm is RSA-240, which has 240 decimal digits (795 bits) and is the product of two large primes.

Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer. However, current technology can only run this algorithm for very small numbers. Hingga Oktober 2012 the largest number that has been factored by a quantum computer running Shor's algorithm is 21.

Other computational applications

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (2048-bit primes are common). RSA relies on the assumption that it is much easier (that is, more efficient) to perform the multiplication of two (large) numbers $x$ and $y$ than to calculate $x$ and $y$ (assumed coprime) if only the product $xy$ is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation (computing $a^{b}{\bmod {c}}$), while the reverse operation (the discrete logarithm) is thought to be a hard problem.

Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers. Carter and Wegman generalized this method to $k$-independent hashing by using higher-degree polynomials, again modulo large primes. As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that the probe sequence covers the whole table.

Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits. Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than $2^{16}$. Prime numbers are also used in pseudorandom number generators including linear congruential generators and the Mersenne Twister.

Other applications

Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including abstract algebra and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line, or so that every triangle formed by three of the points has large area. Another example is Eisenstein's criterion, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square.

The connected sum of two prime knots

The concept of a prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a finite field with a prime number of elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. The prime decomposition of 3-manifolds is another example of this type.

Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics, and have been used metaphorically in the arts and literature. They have also been used in evolutionary biology to explain the life cycles of cicadas.

Constructible polygons and polygon partitions

Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a Fermat prime.

Fermat primes are primes of the form

$F_{k}=2^{2^{k}}+1,$

with $k$ a nonnegative integer. They are named after Pierre de Fermat, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, but $F_{5}$ is composite and so are all other Fermat numbers that have been verified as of 2017. A regular $n$-gon is constructible using straightedge and compass if and only if the odd prime factors of $n$ (if any) are distinct Fermat primes. Likewise, a regular $n$-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of $n$ are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form $2^{a}3^{b}+1$.

It is possible to partition any convex polygon into $n$ smaller convex polygons of equal area and equal perimeter, when $n$ is a power of a prime number, but this is not known for other values of $n$.

Quantum mechanics

Beginning with the work of Hugh Montgomery and Freeman Dyson in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of quantum systems. Prime numbers are also significant in quantum information science, thanks to mathematical structures such as mutually unbiased bases and symmetric informationally complete positive-operator-valued measures.

Biology

The evolutionary strategy used by cicadas of the genus Magicicada makes use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators from synchronizing with these cycles. In contrast, the multi-year periods between flowering in bamboo plants are hypothesized to be smooth numbers, having only small prime numbers in their factorizations.

Arts and literature

Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as La Nativité du Seigneur (1935) and Quatre études de rythme (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".

In his science fiction novel Contact, scientist Carl Sagan suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975. In the novel The Curious Incident of the Dog in the Night-Time by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with Asperger syndrome. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel The Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.

External links

• Hazewinkel, Michiel, ed. (2001) [1994], "Prime number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 
• Caldwell, Chris, The Prime Pages at primes.utm.edu.
• Prime Numbers di In Our Time di BBC.
• Plus teacher and student package: prime numbers from Plus, the free online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge.

Generators and calculators

• Prime factors calculator can factorize any positive integer up to 20 digits.
• Fast Online primality test with factorization makes use of the Elliptic Curve Method (up to thousand-digits numbers, requires Java).
• Huge database of prime numbers
• Prime Numbers up to 1 trillion