Fermat's little theorem

Muchinyorwa chino, tichatarisa imwe yedzidziso huru mudzidziso ye integers -  Fermat's little theoremakatumidzwa zita remuFrance nyanzvi yemasvomhu Pierre de Fermat. Tichaongororawo muenzaniso wekugadzirisa dambudziko kubatanidza zvinyorwa zvakaratidzwa.

Chirevo che theorem

1. Pakutanga

If p inhamba huru a inhamba isina kupatsanurwa nayo pipapo a^p-1 - 1 rakakamurwa ne p.

Zvakanyorwa zviri pamutemo sezvizvi: a^p-1 ≡ 1 (kupesana p).

Cherechedza: Nhamba yepamusoro inhamba yechisikigo inongopatsanurika neXNUMX uye pachayo pasina chasara.

Semuyenzaniso:

• a = 2
• p = 5
• a^p-1 - 1 = 2^{5 - 1} - 1 = 2⁴ - 1 = 16 - 1 = 15
• nhamba 15 rakakamurwa ne 5 pasina chasara.

2. Zvimwe

If p inhamba huru, a chero nhamba, ipapo a^p kufananidzwa ne a module p.

a^p ≡ a (kupesana p)

Nhoroondo yekutsvaga humbowo

Pierre de Fermat akagadzira theorem muna 1640, asi haana kuzviratidza pachake. Gare gare, izvi zvakaitwa naGottfried Wilhelm Leibniz, muzivi wokuGermany, nyanzvi yezvidzidzo, nyanzvi yemasvomhu, nezvimwewo. Zvinodavirwa kuti akanga atova nouchapupu hwacho pakasvika 1683, kunyange zvazvo husina kumbobudiswa. Zvinokosha kuziva kuti Leibniz akawana theorem pachake, asingazivi kuti yakanga yatogadzirwa kare.

Humbowo hwekutanga hweiyo theorem yakabudiswa muna 1736, uye ndeyeSwitzerland, German uye masvomhu uye makanika, Leonhard Euler. Fermat's Little Theorem inyaya yakakosha ye theorem yaEuler.

Muenzaniso wedambudziko

Tsvaga yasara nhamba 2¹² on 12.

mhinduro

Ngatimbofungidzira nhamba 2¹² as 2-2¹¹.

11 inhamba huru, saka, nediki theorem yaFermat yatinowana:

2¹¹ ≡ 2 (kupesana 11).

Nokudaro, 2-2¹¹ ≡ 4 (kupesana 11).

Saka nhamba 2¹² rakakamurwa ne 12 neimwe yasara yakaenzana ne 4.