Ƙananan ka'idar Fermat

A cikin wannan ɗaba'ar, za mu yi la'akari da ɗaya daga cikin manyan jigogi a cikin ka'idar lamba -  Ƙananan ka'idar Fermatmai suna bayan masanin lissafin Faransa Pierre de Fermat. Za mu kuma yi nazarin misali na warware matsalar don ƙarfafa abin da aka gabatar.

Bayanin ka'idar

1. Farko

If p babban lamba ne a lamba ce wacce ba a raba ta psa'an nan a^p-1 - 1 raba ta p.

An rubuta shi a hukumance kamar haka: a^p-1 1 (da p).

lura: Babban lamba lambar halitta ce wacce ke iya rarrabawa ta XNUMX kawai kuma ita kanta ba tare da saura ba.

Misali:

• a = 2
• p = 5
• a^p-1 - 1 = 2^{5 - 1} - 1 = 2⁴ - 1 = 16 - 1 = 15
• lambar 15 raba ta 5 ba tare da saura ba.

2. Madadin

If p babban lamba ne, a kowace lamba, to a^p m da a mai canji p.

a^p ≡ a (da p)

Tarihin gano shaida

Pierre de Fermat ya tsara ka'idar a cikin 1640, amma bai tabbatar da kansa ba. Daga baya, Gottfried Wilhelm Leibniz, masanin falsafar Jamus, masanin ilimin lissafi, masanin ilimin lissafi, ya yi hakan. An yi imanin cewa ya riga ya sami hujja a shekara ta 1683, kodayake ba a taɓa buga shi ba. Abin lura ne cewa Leibniz ya gano ka'idar da kansa, ba tare da sanin cewa an riga an tsara shi a baya ba.

Tabbacin farko na ka'idar an buga shi a cikin 1736, kuma nasa ne na Swiss, Jamusanci da mathematician da makaniki, Leonhard Euler. Karamin Theorem na Fermat wani lamari ne na musamman na ka'idar Euler.

Misalin matsala

Nemo ragowar lamba 2¹² on 12.

Magani

Bari mu yi tunanin lamba 2¹² as 2⋅2¹¹.

11 babban lamba ce, don haka, ta ƙaramin ka'idar Fermat muna samun:

2¹¹ 2 (da 11).

Saboda haka, 2⋅2¹¹ 4 (da 11).

Don haka lambar 2¹² raba ta 12 tare da saura daidai 4.