Kumumanaʻo Helu Kumu
ʻO nā helu mua kekahi o nā mea maʻalahi loa akā pohihihi loa i ka makemakika. He maʻalahi no ka mea he pololei ko lākou wehewehe ʻana: ʻo ka helu mua he helu holoʻokoʻa nui aʻe ma mua o 1 nona nā mahele maikaʻi ʻelua wale nō: 1 a me ia iho. He mea pohihihi no ka mea, ʻoiai ua aʻo ʻia no nā tausani makahiki, nui nā nīnau koʻikoʻi e pili ana i nā helu mua i pane ʻole ʻia a hiki i kēia lā. ʻO ka lālā o ka makemakika e aʻo ana i nā waiwai, nā ʻano, a me ka hoʻolaha ʻana o nā helu mua e kūkākūkā pinepine ʻia ma ke kumumanaʻo helu, a ua ʻike nui ʻia kēia kumuhana ʻo ke kumumanaʻo helu mua.
Ka Wehewehena a me nā Laʻana o nā Helu Prime
Hoʻomaka nā helu mua me 2, 3, 5, 7, 11, 13, 17, 19, a pēlā aku. Mai kēia papa inoa, hiki iā kākou ke ʻike ʻo 2 wale nō ka helu mua like. Hiki ke puʻunaue ʻia nā helu like ʻē aʻe e 2, no laila he ʻoi aku ma mua o ʻelua mau mahele a ʻaʻole hiki ke kapa ʻia he mua.
ʻO ke kū'ē o ka helu mua he helu hui, ʻo ia hoʻi kahi helu piha i ʻoi aku ma mua o 1 nona nā mahele ʻelua. No ka laʻana, he 4 nā mahele 1, 2, a me 4; he 12 nā mahele 1, 2, 3, 4, 6, a me 12. ʻAʻole mua a hui pū ʻole ka helu 1, no ka mea, hoʻokahi wale nō mahele ona, ʻo 1.
He mea nui ka hoʻomaopopo ʻana i nā helu prime no ka mea ua kapa pinepine ʻia lākou ʻo nā "poloka kūkulu" o nā helu piha. Hiki ke hoʻokaʻawale ʻia kēlā me kēia helu piha i ʻoi aku ma mua o 1 i kahi huahana o nā helu prime.
Nā Manaʻo Kumu o ka Arithmetic
ʻO kekahi o nā hopena koʻikoʻi loa i ke kumumanaʻo helu mua ʻo ia ka Fundamental Theorem of Arithmetic. Ke ʻōlelo nei kēia theorem e hiki ke kākau ʻia kēlā me kēia helu piha i ʻoi aku ma mua o 1 ma ke ʻano he huahana kū hoʻokahi o nā helu mua, a hiki i ke kaʻina o kāna mau kumu.
ʻo kahi laʻana:
– 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
– 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7
ʻO kēia ʻano kū hoʻokahi o ka factorization mua ke kumu no nā manaʻo makemakika he nui, mai ka mahele ʻana, ka liʻiliʻi loa o ka maʻamau (LCM), ka factor maʻamau nui loa (GCF), a i nā algorithms like ʻole i ka ʻepekema kamepiula.
Pehea e hoʻoholo ai i nā helu mua
Aia kekahi mau ʻano hana no ka hoʻoholo ʻana inā he helu mua kahi helu. ʻO ke ʻano hana maʻalahi loa, ʻo ia ka nānā ʻana inā hiki ke puʻunaue ʻia ka helu e kekahi helu ʻē aʻe ma mua o 1 a me ia iho. Eia nō naʻe, ʻaʻole pono kēia ʻano hana no nā helu nui.
ʻO kahi ʻano hana ʻoi aku ka maikaʻi, ʻo ia ke nānā i nā mahele a hiki i ke kumu huinaha o ka helu. Inā loaʻa i kahi helu n nā kumu ʻē aʻe ma mua o 1 a me n, a laila ʻaʻole i ʻoi aku kekahi o nā kumu ma mua o √n. No ka laʻana, no ka nānā ʻana inā he prime ʻo 97, e hoʻāʻo wale iā ia no ka mahele ʻana e nā helu prime ≤ √97 (ma kahi o 9,8), ʻo ia hoʻi 2, 3, 5, a me 7. ʻOiai ʻaʻole hiki ke mahele ʻia ʻo 97 e kekahi o kēia mau ʻehā, he prime ia.
Eia kekahi, aia nā ʻano hana kuʻuna e like me ka Sieve of Eratosthenes, he mea kūpono no ka loaʻa ʻana o nā helu mua a hiki i kahi palena. Pili ke ʻano hana i ka papa inoa ʻana i nā helu āpau mai 2 a i N, a laila e hōʻailona ana i nā multiples o 2, nā multiples o 3, nā multiples o 5, a pēlā aku. ʻO nā helu i hōʻailona ʻole ʻia he primes.
Nā ʻAno Hoʻohālikelike a me ka Hoʻolaha ʻana o nā Helu Prime
ʻO kekahi o nā mea e hoʻolilo ai i nā helu mua i mea hoihoi, ʻo ia ko lākou hoʻolaha like ʻole. ʻAʻohe ʻano maʻalahi e hoʻopuka koke i nā helu mua āpau. I kekahi manawa ʻike ʻia nā helu mua e kokoke ana kekahi i kekahi, e like me 11 a me 13, a i ʻole 17 a me 19, akā hiki i ka mamao ma waena o lākou ke nui aʻe.
ʻO kahi nīnau koʻikoʻi i ke kumumanaʻo helu mua: "ʻEhia pinepine ka loaʻa ʻana o nā helu mua?" No ka pane ʻana i kēia, ua hoʻomohala nā makemakika i ka Prime Number Theorem. Ke ʻōlelo nei kēia theorem ʻo ka helu o nā primes i emi iho a like paha me n, i kākau ʻia ʻo π(n), ua like ia me n / ln(n). ʻO ke ʻano kēia, i ka piʻi ʻana o n, lilo nā primes i mea kakaikahi, akā ke hoʻomau nei lākou i ka hiki ʻole ke pau.
Eia kekahi laʻana, no ka n nui loa, ʻo ka likelika e lilo kahi helu kaulele a puni ʻo n i prime ma kahi o 1/ln(n). Hāʻawi kēia i kahi ʻike helu o ka hoʻolaha ʻana o nā helu prime.
Nā Helu Māhoe Prime a me nā Kuhiakau Kaulana
ʻO nā māhoe primes he mau hui o nā helu prime nona ka ʻokoʻa he 2, e like me (3, 5), (11, 13), (17, 19), a pēlā aku. ʻO kahi nīnau nui e waiho nei i kēia lā ʻo ia ka Twin Prime Conjecture: "He nui loa nā hui o nā māhoe primes?" ʻOiai he nui nā hōʻike kākoʻo a me ka holomua koʻikoʻi i nā makahiki i hala iho nei, ʻaʻole i hōʻoia piha ʻia kēia kuhi.
Eia kekahi, aia ka Goldbach Conjecture, kahi e ʻōlelo nei hiki ke kākau ʻia kēlā me kēia helu like ma mua o 2 ma ke ʻano he huina o ʻelua mau helu mua. Eia kekahi laʻana:
– 10 = 3 + 7
– 28 = 11 + 17
– 100 = 47 + 53
Ua hoʻāʻo ʻia kēia kuhi no nā helu nui loa a he ʻoiaʻiʻo mau ia ma lalo o kēia mau hoʻāʻo, akā ʻaʻohe hōʻike piha piha.
Ke Kūlana o nā Helu Prime i ka Cryptography Hou
Ma waho aʻe o ka makemakika maʻemaʻe, he kuleana koʻikoʻi ko nā helu mua i ke ola hou, ʻoiai hoʻi i ka palekana ʻike. ʻO ka ʻōnaehana crypto RSA, kekahi o nā ʻano hoʻopāʻālua kaulana loa, e hilinaʻi ana i ka paʻakikī o ka hoʻokaʻawale ʻana i nā helu nui i nā kumu mua. Ma RSA, hiki ke hana ʻia kahi kī lehulehu mai ka huahana o ʻelua mau helu mua nui. ʻO ka hoʻemi ʻana i kēia huahana i kāna mau helu mua ʻelua he pilikia paʻakikī loa inā lawa ka nui o nā helu, no laila e hāʻawi ana i ke kumu no ka palekana.
I nā huaʻōlelo ʻē aʻe, ʻo ka nani o nā helu mua ʻaʻole wale ia he kumumanaʻo akā he hana pono hoʻi, ʻoiai lākou e kōkua ana i ka mālama ʻana i ka palekana o nā hana kikohoʻe, nā kamaʻilio i hoʻopāʻālua ʻia, a me ka hoʻololi ʻana o ka ʻikepili pilikino ma ka pūnaewele.
Pani
ʻO ke kumumanaʻo helu mua kekahi o nā kahua waiwai loa i ka makemakika. Mai ka wehewehe maʻalahi o kahi helu mua, hiki mākou i nā manaʻo hohonu e like me ka factorization kū hoʻokahi, ka theorem hoʻolaha mua, a me nā kuhi i hoʻoholo ʻole ʻia. Ke hoʻomau nei nā helu mua i ka hoihoi iā mākou no ka mea ke kau nei lākou ma ka palena ma waena o ke kauoha a me ka randomness: me he mea lā ʻaʻohe ʻano, akā e mālama ana i nā kānāwai makemakika hohonu.
ʻAʻole wale ke aʻo ʻana i nā helu prime e pili ana i ka ʻike ʻana i nā helu prime a ʻo wai ka mea ʻaʻole, akā e pili ana hoʻi i ka hoʻomaopopo ʻana pehea lākou e "hoʻokaʻawale ʻia" ai i waena o nā helu piha, no ke aha lākou e pāʻani ai i kahi kuleana koʻikoʻi, a pehea lākou e hoʻohuli ai i ka ʻenehana hou. No kēia kumu, ʻo ke kumumanaʻo helu prime kekahi o nā kumuhana i noiʻi nui ʻia a hoihoi hoʻi no nā haumāna makemakika, nā kumu, a me nā mea noiʻi a hiki i kēia lā.
