MATSALA TA GOMA TA HILBERT A KAN LAMBOBIN RABO: ABUBUWAN DA BA A SANI BA 86 SUN ISA

Matsala ta goma ta Hilbert, wadda aka gabatar a 1900, tana neman algorithm da zai ɗauki kowace lissafin polynomial mai ma’aunai na lambobi cikakku, P(x₁, …, xₙ) = 0, ya yanke hukunci ko tana da mafita a lambobi cikakku. A 1970, Matiyasevich, yana gina a kan aikin Davis, Putnam da Robinson daga 1961, ya tabbatar cewa babu irin wannan algorithm. Matsalar ba ta da hanyar yanke hukunci (undecidable).

Masana lissafi sai suka fara ƙirgawa. Abubuwa nawa da ba a sani ba lissafi ke buƙata kafin tambayar ta zama marar hanyar yanke hukunci? Zhi-Wei Sun, a Jami’ar Nanjing, ya nuna cewa 11 sun isa ga lambobi cikakku, kuma kwanan nan ya saukar da shi zuwa 10.

Tambaya ɗaya, ga lambobin rabo

Yi tambayar ga lambobin rabo (rational numbers) — fractions — maimakon haka. Hanyar gargajiya za ta bi ta lambobi cikakku: idan za ka iya ware lambobi cikakku daga cikin lambobin rabo ta amfani da lissafin polynomial, ka’idar Matiyasevich za ta ci gaba da aiki. Takardar ta jera matakan. Robinson ya ba da ma’anar ma’ana (logical definition) ta lambobi cikakku a cikin lambobin rabo a 1949, Poonen ya sauƙaƙa ta a 2009. Koenigsmann ya nuna a 2016 cewa za a iya siffanta lambobin da ba cikakku ba ta irin wannan lissafi, kuma aikin da ya biyo baya ya rage adadin abubuwan da ba a sani ba da ake buƙata don haka zuwa 10 (Daans, 2024), sannan 7 (Sun).

Sannan, takardar ta ce, “kwanan nan sosai, OpenAI ya fitar da preprint” wanda ya tabbatar cewa matsala ta goma ta Hilbert a kan lambobin rabo ba ta da hanyar yanke hukunci. Wannan preprint, mai kwanan wata 6 ga Oktoba 2026 kuma aka samar da shi a wani ma’ajiyar GitHub ta OpenAI, ya dogara ne kan abin da Sun ke kira “ƙiyasin tsawo na mahaɗa biyar” (five-contact height estimate) da kuma wani gini da ke kawar da “odd poles”. Yadda aka samar da shi ba a bayyana shi a cikin takardar Sun ba.

Ka’ida ta 1.1

Gudummawar Sun lamba ce. Ka’ida ta 1.1: HTP₈₆(ℚ) ba ta da hanyar yanke hukunci. A kalmomi: babu algorithm da ke yanke hukunci, ga kowace polynomial f(x₁, …, x₈₆) mai ma’aunai na lambobi cikakku, ko f = 0 tana da mafita a lambobin rabo.

Hujjar tana shigar da nasa sakamakon na abubuwa 10 da ba a sani ba ga lambobi cikakku cikin injin preprint ɗin OpenAI. Ɗigo goma a kan wata tsayayyar lanƙwasar elliptic — a nan y² = x³ − d₀²x a kan lambobin a + b√2, inda d₀ = 75 − 53√2 — suna lulluɓe lambobi cikakku goma. Lambobin firamare na musamman, waɗanda ƙiyasin tsawon yake gano su, suna bar lissafin ya duba ko polynomial ɗin da aka shigar yana zama sifili a waɗannan lambobi goma. Jerin “masu gadi” (guards) mai iyaka yana kawar da mafitan ƙarya da suka rage. Idan aka gudanar da shi tare da bincike mai sauƙi na mafita a lambobi cikakku, wannan yana mayar da kowane oracle na lambobin rabo zuwa hanyar yanke hukunci ga lambobi cikakku — wadda ba za ta iya wanzuwa ba.

Ajiye abubuwan da ba a sani ba

Mafi yawan shafuka 42 an keɓe su ne ga tattalin arziki. Sun ya jera dabaru biyar. Canjin daidaitawa (coordinates) yana motsa “mahaɗa” biyar na ƙiyasin tsawon zuwa ɗigo masu iyaka. Maɓalli guda ɗaya na “gangare” (slope) yana yi wa bincike ashirin da ɗaya hidima a lokaci guda. Rukuni ɗaya na ɗigo goma sha ɗaya da ake rabawa a kan lanƙwasar yana maye gurbin da yawa daban-daban — wanda ke buƙatar gina iyalan lambobin firamare gaba ɗaya waɗanda suka dace da juna, ta amfani da ka’idar saura ta China (Chinese remainder theorem) da aka faɗaɗa da sakamakon Tao da Ziegler kan tsarin polynomial a cikin lambobin firamare, da sauransu. An sake tsara binciken ta yadda sharuɗɗa 33 suke maye gurbin 42. Kuma a ƙarshe, kowane zaɓi na algebra da ya rage da kowane sharaɗin “wannan ba dole ya zama sifili ba” an naɗe su cikin maɓalli ɗaya na ƙarshe.

Ƙirgar, kamar yadda aka rubuta ta a cikin takardar:

2(10 + 11) + 1 + 2 · 21 + 1 = 86

— daidaitawa biyu ga kowane ɗaya daga cikin ɗigo 21 a kan lanƙwasar, gangare ɗaya da ake rabawa, abubuwa biyu masu taimako da ba a sani ba ga kowane ɗaya daga cikin bincike 21, da maɓalli ɗaya na ƙarshe wanda ke shanye sauran.

Ba kalma ta ƙarshe ba

Sun ya fito fili cewa 86 “iyaka ta sama ce da wannan gini yake bayarwa”, ba iƙirari ba cewa ƙananan abubuwan da ba a sani ba za su gaza. A kan lambobi cikakku, adadin ya tafi daga 11 zuwa 10; a kan lambobin rabo, tseren ya fara ne yanzu.

Sakamakon kuma yana dogara ne kan aikin kwanan nan wanda ba a yi masa bita na takwarorin aiki ba, a cewar manazarta: preprint ɗin OpenAI daga farkon Oktoba da takardar Sun ta kansa ta abubuwa 10 da ba a sani ba, wadda aka sanya a arXiv a wannan shekara. Takardar Sun, ita ma, preprint ce. Shafukanta 42 na ilimin lambobi — sieves, tsarin lambobin firamare da lanƙwasar elliptic — yanzu wasu masana lissafi za su duba su.