Te kaupapa iti a Fermat

I roto i tenei whakaputanga, ka whakaarohia e matou tetahi o nga kaupapa matua o te ariā o te tauoti -  Te kaupapa iti a Fermati tapaina ki te ingoa o te tohunga pangarau French a Pierre de Fermat. Ka wetewetehia e matou tetahi tauira o te whakaoti rapanga hei whakakotahi i nga korero kua whakaatuhia.

Tauākī o te ariā

1. Tuatahi

If p he tau pirimia a he tauoti e kore e whakawehea e pka a^wh-1 - 1 Wehea e te p.

He penei te tuhi okawa: a^wh-1 ≡ 1 (ki te p).

Tuhipoka: Ko te tau pirimia he tau maori ka wehea e te XNUMX me te kore e toe.

Hei tauira:

• a = 2
• p = 5
• a^wh-1 - 1 = 2^{5 - 1} - 1 = 2⁴ – 1 = 16 – 1 = 15
• tau 15 Wehea e te 5 kahore he toenga.

2. Whakakoatanga

If p he tau matua, a tetahi tauoti, katahi a^p rite ki a modulo p.

a^p ≡ a (ki te p)

Te hitori o te kimi taunakitanga

I hangaia e Pierre de Fermat te kaupapa i te tau 1640, engari kaore i whakamatauhia e ia. I muri mai, na Gottfried Wilhelm Leibniz, he tohunga mohio Tiamana, tohunga arorau, tohunga pangarau, aha atu. E whakaponohia ana kua riro i a ia te tohu i mua i te tau 1683, ahakoa kaore i panuitia. Ko te mea nui kua kitea e Leibniz te kaupapa, me te kore e mohio kua oti kee te whakatakoto i mua.

Ko te tohu tuatahi o te kaupapa i whakaputaina i te tau 1736, a no te Swiss, Tiamana me te tohunga pangarau me te miihini, a Leonhard Euler. He take motuhake a Fermat's Little Theorem o Euler's theorem.

He tauira o te raruraru

Kimihia te toenga o te tau 2¹² on 12.

otinga

Me whakaaro tatou he tau 2¹² as 2⋅2¹¹.

11 he tau pirimia, no reira, na te kaupapa iti a Fermat ka whiwhi tatou:

2¹¹ ≡ 2 (ki te 11).

No reira, 2⋅2¹¹ ≡ 4 (ki te 11).

Na te tau 2¹² Wehea e te 12 me te toenga rite ki 4.