Ezinye izinkinga zezibalo zakhiwe kalula kangangokuthi zingachazwa enganeni, kodwa zinzima kangangokuthi zithatha izizukulwane zezazi zezibalo. Enye inkinga enjalo yilokho okubizwa ngokuthi inkinga yebanga leyunithi: Beka amaphuzu \(n\) endizeni. Zingaki izibhangqwana zamaphuzu ezingaba nebanga elingu \(1\) ngqo? Inkinga isukela kuPaul Erdős futhi ifundwe kusukela ngo-1946. I-OpenAI manje isishicilele ukuthi imodeli yangaphakathi iphikise ukuqagela osekunesikhathi eside mayelana nale nkinga.
Uma uqala ukubona, umbuzo uzwakala ungenangozi. Uma ubeka amaphuzu \(n\) emgqeni oqondile, uthola cishe izikhawu zobude obungu \(1\) \(n-1\) . Uma uhlela amaphuzu njengegridi, njengasephepheni legrafu, uthola okuningi kakhulu kwalezi zikhawu: ngokuvundlile nangokuqondile phakathi kwamaphuzu aseduze. U-Erdős wayesethole ukwakheka okungcono kancane kunomugqa. Nokho, isikhathi eside kwakukholelwa ukuthi lokhu akunakuthuthukiswa kakhulu. Ngokomthetho, umcabango wawuwukuthi inani eliphezulu lezikhawu zamayunithi anjalo likhula cishe cishe ku \(n^{1+o(1)}\) , okungukuthi, ngokushesha kancane kune \(n\) , kodwa hhayi nge-exponent eyengeziwe ehleliwe.
Leli yiphuzu elimangalisayo impela: Imodeli ye-OpenAI (engadalulwanga) yakha hhayi isibonelo esisodwa esiphikisayo, kodwa umndeni ongenamkhawulo wamasethi wamaphuzu \(P\) lapho inani lamabanga amayunithi okungenani liyi \(|P|^{1+\delta}\) , ene- \(\delta>0\) ehleliwe. Ukulungiswa kamuva nguWill Sawin, ngokusho kwe-OpenAI, kuze kuveze \(\delta=0{,}014\) . Lokhu kuzwakala kuncane, kodwa kukhulu ngokwezibalo: Akuseyona insalela ye-logarithmic, kodwa inzuzo yangempela ye-polynomial.
Isibonelo esilula sibonisa umqondo oyisisekelo. Ake sicabangele inombolo eyinkimbinkimbi
\[
u=\frac{2+i}{2-i}=\frac{3+4i}{5}=\frac35+\frac45i.
\]
Le nombolo inenani eliphelele elingu- \(1\) , ngoba
\[
\left|\frac35+\frac45i\right|=\sqrt{\left(\frac35\right)^2+\left(\frac45\right)^2}=1.
\]
Ngakho-ke, uma \(x\) iyiphuzu endizeni, khona-ke \(x\) kanye ne \(x+u\) zihlukene ncamashi \(1\) . Ngokukhethekile, isibonelo, amaphuzu atholakala ku-...
\[
0
\quad\text{und}\quad
\frac35+\frac45i
\]
iyunithi eyodwa kuphela ehlukene. Ngakho-ke inkulumo yethiyori enani iveza isiqondiso sejiyometri sobude \(1\) . Lokhu akukabi yisiqinisekiso esikhulu sokuphikisana, kodwa inguqulo encane yale ndlela: umuntu akabheki nje kuphela amabanga eyunithi avundlile naqondile njengakugridi, kodwa neziqondiso eziningi ezikhiqizwe ngokwezibalo, zonke ezinobude obuqondile \(1\) .
Ukwakhiwa kwangempela kusebenzisa amathuluzi anamandla kakhulu. Esikhundleni sokusebenza kuphela ngezinombolo eziphelele ze-Gaussian \(\mathbb{Z}[i]\) , ubufakazi busebenzisa amasimu ezinombolo ze-algebraic ayinkimbinkimbi kakhulu \(K=L(i)\) . Kukhona izakhi eziningi zefomu
\[
u=\frac{\alpha}{c(\alpha)}
\]
yakhiwe, lapho \(c\) idlala indima yokuhlanganiswa okuyinkimbinkimbi. Umphumela obalulekile uwukuthi: phakathi kokushumeka okuyinkimbinkimbi okufanele, lokhu \(u\) ngakunye kunobukhulu obungu- \(1\) . Ngakho-ke bangabameleli beziqondiso eziningi ezahlukene zamayunithi.
Ngamagama aqatha kakhulu, isibonelo sangempela esiphikisayo asibukeki njengesithombe esincane, esihle esinamachashazi ayishumi, kodwa sifana neqembu elikhulu lamachashazi akhiwe ngokwezibalo. Uthatha igridi enobukhulu obuphezulu, usike amaphuzu afanele kuyo, bese uwabuyisela endaweni evamile. Ekubhalweni kobufakazi, lokhu kuvezwa cishe ngesimo...
\[
P_j=\pi_1\big((y+\Lambda_j)\cap W\big),
\]
Ngamanye amazwi: Umuntu uthatha amaphuzu kugridi eshintshiwe \(y+\Lambda_j\) , awakhawulele ngesifunda \(W\) , bese ewafaka nge \(\pi_1\) ku-coordinate eyinkimbinkimbi, okungukuthi, ku- \(\mathbb{C}\cong\mathbb{R}^2\) . Umehluko omningi phakathi kwala maphuzu ube yizinto eziqondile \(u\) ezinobukhulu \(1\) . Ngakho-ke, ngemva kokuma, aba amabanga eyunithi yangempela endizeni.
Isifaniso esibonakalayo yilesi: Igridi ejwayelekile isebenzisa izikhombisi-ndlela ezimbalwa ezilula, njengokunene, kwesobunxele, phezulu, naphansi. Nokho, ukwakhiwa okusha kukhiqiza inani elikhulu lezikhombisi-ndlela ezifihliwe ezithathwe ku-theory yezinombolo ze-algebraic. Isiqondiso ngasinye siyunithi eyodwa ncamashi ubude. Ngenxa yokuthi kunezikhombisi-ndlela eziningi kangaka futhi amaphuzu amaningi ahambisana nazo, amabanga eyunithi iyonke makhulu kunalokho okucatshangelwayo okudala.
Kuyaphawuleka futhi ukuthi ubufakazi buvelaphi. Ngokusho kwe-OpenAI, ikhambi latholakala ngokuzimela ngemodeli yokucabanga evamile, hhayi ngohlelo lwezibalo oluqeqeshwe ngqo le nkinga. Ubufakazi babuyekezwa ngaphakathi nangaphandle futhi bahunyushwa ngendlela efundeka kalula. Amanothi ahambisanayo avela kososayensi bezibalo bangaphandle agcizelela nokuthi lokhu akuyona nje inguqulo ezenzakalelayo yendlela eyaziwayo, kodwa ukuxhumana okungalindelekile phakathi kwe-geometry ehlukene kanye ne-algebraic number theory.
Mhlawumbe lesi yisona sizathu sangempela esenza lo mphumela ube mnandi kangaka. Akukhona nje ukuxazulula inkinga ngendlela efanele nge-AI. Kumayelana nokuthola indlela eyayingabonakali kubantu abaningi: ukubhekana nenkinga yejiyometri mayelana namabanga endizeni ngamathuluzi ajulile avela ku-theory yezinombolo. Lokhu akwenzi izibalo zibe yinto engabalulekile kangako. Kodwa kuyinika umphikisi omusha, onamandla angakhululekile.