Nazariyai adadho

Nazariyai adadho, ilmest dar borai adadhoi butun.

Mafhumi adadi butun, inchunin amalhoi arifmetiki bo adadho az davrahoi qadim malum buda, yake az abstraksiya (mafhum)-hoi avvali­ni matematika ba hisob meravad.

Dar bayni adadhoi butup, yane,… —3, —2, —1, 0, 1, 2, 3,… adadhoi naturali — adadhoi butuni musbat 1, 2, 3, …, khosiyati onho va amalho bo onho maqomi khos dorad. Hamai adad­hoi naturalii az yak kalon ba du sinf judo meshavand: ba sinfi yakum adadhoi naturalii faqat du taqsimkunaidai naturali (1 va khudi hamon adad) doshta va ba sinfi duyum hamai adadhoi naturalin boqimonda mansuband. Adadhoi sinfi yakum adadhoi sodda, adadhoi sinfi duyum adadhoi tarkibi nom giriftaand. Khosiyati adadhoi sodda va aloqai onhoro bo digar adadhoi naturali Evklid (asri 3 to melod) omukhtaast. Agar hamai adadhoi naturaliro pai ham navisem, zichii nisbii adadhoi sodda dar in qator toraft kam meshavad: shumorai onho dar 10 adadi avvali qator 4-to, yane 40%; dar 100 adad 25-to, yane 25%; baroi 1000 adad 168-to, yane 17%; baroi million adid 78 498-to, yane 8% va gayra, ammo ba in nigoh nakarda miqdori onho beintihost. Dar bayni adadhoi sodda jufti adadhoe vome- khurand, ki farqashon ba 2 k barobar ast (onhoro «jufthoi sodda» — «kekizakho» menomand). Bointiho va beintiho budani miqdori in guna adadho to hol isbot nashudaast. Joygirshavii adadhoi sodda dar qatori adadhoi naturali yake az masalahoi avvalini nazariyai adadhoi sod­da ba nisob meraft. Omuzishi in masala boisi ba vujud omadani al­goritm (qoida)-e gardid, ki baroi tartib dodani jadvali adadhoi sod­da imkoniyat dod (nig. Galberi Era­tosfen). Evklid dar «Usul» nom asarash qoidai hisob kardani ka- lontarin taqsimkunandai du adad (algoritmi Evklid) -ro nishon dod, ki az on dar borai ba zarbshavandanoi sodda yakqimata judo, namudani adadhoi naturali teoremae barmeoyad. Diofant dar «Arifmetika» nom asarash nazariyai muodilahoya koeffisientnoyash butunro muntazam bayon namud (nig. Muodilahoi Dio­fant) ki on ba inkishofi minbadai Nazariya avtomatho takon dod.  P. Ferma (asri 17) dar nazariyai muodilahoi diofanti va nazariyai ba taqsimshavandagii adadho aloqamand kashfiyothoi buzurg ba amal ovard (nig. Teoremai buzurgi Ferma). Tadqiqoti Fermaro doir ba taqsimshavandagii adadho davom doda, L. Eyler teoremaero isbot kard, ki tamimi ba nom teo­remai khurdi Ferma gardid. Ba nom teoremai buzurgi Fermaro niz ba­roi p = 3 Eyler isbot namudaast.

L. Eyler usulhoi tahlili matematikiro baroi halli masalahoi Nazariyai avtomatho istifoda burd. Dar natija me­todi funksiyahoi hamzarbi Eyler, metodi doiravii Khardi-Litlvud va nihoyat metodi summahoi trigono­metrii I. .M. Vinogradov ba vujud omad; bo yorii in metod u yak qator masalahoi muhimmi Nazariyai avtomathoro hal kard. D. Eyler’teoremaero dar borai beintihoii adadhoi soddai Evklid bo usuli nav isbot namud, ki on sonitar asosi nazariyai dzeta-funksiyaho gardid. Guzorishi nakhustin masalahoi additivi (yane masalahoi bo amali jam aloqamand) bo adadhoi sodda ba L. Eyler va X. Goldbakh mansub ast. Dar solhoi 50- umi asri 19 Nazariyai avtomatho asosan bunyod gar­did, ki bo nomi K. Gauss, J. Lag­ranj, A. Lejandr, P. Dirikhle, P. L. Chebishev, J. Liuvill va digaron aloqai mustahkam dorad. Masalan, Gauss nazariyai muqoisaro ba vujud ovard, ki ba vositai on, masalan, teoremai zirin isbot karda shud: «adadi sodda dar namoya mavrid summai du kvad­rat shuda metavonad, ki agar on namudi 4p+ 1 doshta boshad».

Chebishev boshad, bori avval buzur­gii ziyodshavii funksiyai π(N)-po nishon dod, kb on miqdori adadhoi soddai az N khurd yo ba on barobarro ifoda mekunad.

Dar inkishofi minbadai Nazariyai avtomatho olimoni soveti 10. V. Linnik, A. Ya. Khinchin, N. P. Romanov, A. O. Gelfond, A. I. Vinogradov, A. A. Karatsuba, A. G. Postnikov, B. N. Delone, D. K. Fadeev, A. V. Malishev, N. G. Chudakov, I. P. Kubilyus va digaron sahmi kalon guzoshtaand.

As,: Babaev G., Raspredelenie tse­likh tochek na algebraicheskikh poverkhnos­tyakh, D., 1066; Fayziev R. F., Galberi Eratosfen, umumikuni va tatbiqi on (Adadhoi sodda). D., 1067; Vinog­radov I. M., Osnovi teorii chisel, 9 izd., M., 1082; Osobie varianti meto­da trigonometricheskikh summ. M., 1076; B o r e v i ch  3. P., Sh a f a r e v i ch I. R., Teoriya chisel, 2 izd., M., 1072; K a- r a ts u b a A. A., Osnovi analiticheskoy teorii chisel, M., 1075; Kulikov L. Ya., Algebra i teoriya chisel. M., 1070.