Ithiyori encane kaFermat

Kulolu shicilelo, sizocubungula enye yethiyori eyinhloko kuthiyori yama-integers -  Ithiyori encane kaFermatiqanjwe ngesazi sezibalo saseFrance uPierre de Fermat. Sizophinde sihlaziye isibonelo sokuxazulula inkinga ukuze sihlanganise indaba ethulwayo.

Isitatimende sethiyori

1. Eyokuqala

If p iyinombolo eyinhloko a iyinani eliphelele elingehlukaniseki ngalo pke a^p-1 - 1 zihlukaniswe p.

Ibhalwe ngokusemthethweni kanje: a^p-1 ≡ 1 (ngokumelene p).

Qaphela: Inombolo eyinhloko iyinombolo engokwemvelo ehlukaniseka kuphela ngo-XNUMX futhi yona ngokwayo ngaphandle kokusala.

Ngokwesibonelo:

• a = 2
• p = 5
• a^p-1 - 1 = 2^{5 - 1} - 1 = 2⁴ - 1 = 16 - 1 = 15
• inombolo 15 zihlukaniswe 5 ngaphandle kokusala.

2. Okuhlukile

If p inombolo eyinhloko, a noma iyiphi inombolo ke a^p ukuqhathanisa ne a module p.

a^p ≡ a (ngokumelene p)

Umlando wokuthola ubufakazi

U-Pierre de Fermat wenza i-theorem ngo-1640, kodwa akazange afakazele ngokwakhe. Kamuva, lokhu kwenziwa uGottfried Wilhelm Leibniz, isazi sefilosofi saseJalimane, isazi sezibalo, njll. Kukholakala ukuthi wayesenabo ubufakazi ngo-1683, nakuba bungakaze bushicilelwe. Kuyaphawuleka ukuthi uLeibniz wathola le theory ngokwakhe, engazi ukuthi yayisivele yakhiwe ngaphambili.

The first proof of the theorem was published in 1736, and it belongs to the Swiss, German and mathematician and mechanic, Leonhard Euler. Fermat’s Little Theorem is a special case of Euler’s theorem.

Isibonelo senkinga

Thola inombolo esele 2¹² on 12.

Isixazululo

Ake sicabange inombolo 2¹² as 2-2¹¹.

11 inombolo eyinhloko, ngakho-ke, ngethiyori encane kaFermat esiyitholayo:

2¹¹ ≡ 2 (ngokumelene 11).

Ngakho, 2-2¹¹ ≡ 4 (ngokumelene 11).

Ngakho inombolo 2¹² zihlukaniswe 12 nensalela elingana ne 4.