ʻO ka haʻi liʻiliʻi a Fermat

Ma kēia hoʻolaha, e noʻonoʻo mākou i kekahi o nā theorems nui i ka ʻepekema o integers -  ʻO ka haʻi liʻiliʻi a Fermatua kapa ʻia ma hope o ka mea makemakika Farani ʻo Pierre de Fermat. E nānā pū mākou i kahi laʻana o ka hoʻoponopono ʻana i ka pilikia e hoʻohui i nā mea i hōʻike ʻia.

ʻO ka ʻōlelo o ka theorem

1. Kumumua

If p he huina nui a he integer hiki ole ke puunaueia e palaila, a^p-1 - 1 maheleia e p.

Ua kākau ʻia e like me kēia: a^p-1 ≡ 1 (kūʻē p).

'Ōlelo Aʻo: ʻO ka helu prime he helu kūlohelohe ia e puʻunaue wale ʻia e XNUMX a iā ia iho me ke koena ʻole.

ʻo kahi laʻana:

• a = 2
• p = 5
• a^p-1 - 1 = 2^{5 - 1} - 1 = 2⁴ – 1 = 16 – 1 = 15
• helu 15 maheleia e 5 me ke koena ole.

2. Nā pono ʻē aʻe

If p he helu prima, a kekahi huinahelu, alaila a^p hoʻohālikelike ʻia i a māhele p.

a^p ≡ a (kūʻē p)

Moolelo o ka loaa ana o na hoike

Ua hoʻokumu ʻo Pierre de Fermat i ka theorem i ka makahiki 1640, ʻaʻole naʻe i hōʻoia iā ia iho. Ma hope mai, ua hana ʻia kēia e Gottfried Wilhelm Leibniz, he kanaka kālaiʻike Kelemania, loea, makemakika, etc. He mea mahalo ia ua ʻike ʻo Leibniz i ka theorem iā ia iho, me ka ʻike ʻole ua hana ʻia ma mua.

Ua paʻi ʻia ka hōʻike mua o ka theorem i ka makahiki 1736, a no ka Swiss, German a me ka makemakika a me ka mechanic, ʻo Leonhard Euler. ʻO Fermat's Little Theorem kahi hihia kūikawā o ka theorem a Euler.

Laʻana o kahi pilikia

E huli i ke koena o ka helu 2¹² on 12.

pāʻoihana

E noʻonoʻo kākou i kahi helu 2¹² as 2⋅2¹¹.

11 He helu prima, no laila, ma ka Fermat theorem, loaʻa iā mākou:

2¹¹ ≡ 2 (kūʻē 11).

No laila, 2⋅2¹¹ ≡ 4 (kūʻē 11).

No laila ka helu 2¹² maheleia e 12 me ke koena like me 4.