RUFI GA A + B = C, DA MA'AUNIN ƘARFI 24

Rubuta a + b = c da cikakkun lambobi uku waɗanda ba su da wani abin raba su (factor) na bai ɗaya. Yanzu jera lambobin firamare (prime numbers) da ke raba a, b ko c, sannan ka ninka su tare, kowanne sau ɗaya kawai. Kira sakamakon N.

Mafi yawan lokaci, N ya fi c girma nesa ba kusa ba. Ɗauki 2 + 3 = 5: lambobin firamaren su ne 2, 3 da 5, don haka N = 30, sama da 5 nesa. Amma wani lokaci ana gina lambobin daga ’yan lambobin firamare kaɗan ƙwarai, ƙanana ƙwarai. A 1 + 8 = 9, lambobin firamaren kaɗai su ne 2 da 3, don haka N = 6 — ƙasa da c. A 1 + 80 = 81, inda 80 = 2⁴ × 5 da 81 = 3⁴, N shi ne 30, yayin da c shi ne 81.

Ƙananan lambobin firamare, manyan jimloli?

Har ina wannan zai kai? Shin c zai iya zama mai girma matuƙa yayin da N ke ƙarami? Hasashen abc (abc conjecture) na Masser da Oesterlé ya ce ba zai yiwu ba: c yana zama ƙasa da wani ƙarfi (power) na N. An san cewa wasu maganganu da dama game da cikakkun lambobi suna biyo baya daga gare shi.

Wani preprint da aka wallafa a ranar 3 ga Nuwamba 2026, na Christophe Soulé, na CNRS da Institut des hautes études scientifiques (IHÉS) a Bures-sur-Yvette, Faransa, yana sanar da hujjar wani nau’i mai rauni (weak form) nasa:

c ≤ A(ε) · N^(24+ε)

Anan ε kowace lamba ce mai kyau (positive), ƙanƙanta yadda ka so, A(ε) kuma ƙayyadadden adadi (constant) ne da ya dogara gare ta kaɗai. A kalmomi: c ba zai taɓa wuce wani tsayayyen ninki na N da ƙarfin kusan 24 ba. Ga N = 30, wannan rufi yana kusan 3 × 10³⁵ sau wani ƙayyadadden adadi — mai girma matuƙa, amma tsayayye.

A cewar mai rubuta takardar, a fili, wannan ba shi ne hasashen kansa ba: dokar abc tana biyo baya a nan “ko da yake da ma’aunin ƙarfi na kuskure”, kamar yadda takardar ta faɗa.

Zagayawa ta hanyar lanƙwasai (curves)

Hujjar tana bin hanyar zagayawa. Babban sakamakonta ya shafi lanƙwasai na genus 2 (curves of genus 2) — lissafai (equations) masu abubuwan da ba a sani ba (unknowns) guda biyu, masu wani matakin sarƙaƙiya na musamman. Ga irin wannan lanƙwasa, takardar ta sanya iyaka ga “tsayi” (height) na mafitarta (solutions), wato ma’aunin yadda lambobinsu suke da girma idan an rubuta su, da 12 + ε sau wani adadi da ke haɗe da tsarin lambobi (number system) da mafitar ke cikinsa. Wani hasashe mafi buri na Vojta zai sanya 1 + ε a madadin 12 + ε.

An riga an sani, daga aikin Moret-Bailly, cewa irin wannan iyaka tana nuna (implies) wani nau’i na abc. Parshin, sannan Moret-Bailly, su ne suka gabatar da dabarar kanta. Soulé ne ke aiwatar da ita:

1. Wani gini (construction) na Kodaira yana haɗa wa lanƙwasar genus 2 dukan iyalin lanƙwasai na genus 6 (family of curves of genus 6).
2. Wani adadi da ake auna a kan kowace lanƙwasa ta iyalin, wato “haɗuwa da kanta” ta lissafi (arithmetic self-intersection), ya zama tsayi (height) ne. Babban ka’idar (core theorem) tana sanya masa rufi, da ninki 36 + ε.
3. Don a kai can, takardar ta haɗa ka’idar haɗuwa (intersection theory) ta saman lissafi (arithmetic surfaces), ka’idar Minkowski a kan ƙananan ƙima masu bi-da-bi (successive minima), da lissafai a kan lanƙwasai na genus 3 da 6 da aka yi “tare da taimakon” Voisin.

Daga nan ma’aunin ƙarfi 24 yana fitowa ne daga wata takamaiman lanƙwasa, y² + y = x⁵. Ta hanyar taswirarta (map) ta digiri 5 zuwa kan layi, ninkin 12 ya zama 5/2 × 12 = 30; ƙiyasin wani discriminant sannan yana ninka shi da 4/5, ya ba da 24.

Sakamakon da ke biyo baya, da ƙayyadaddun adadi masu rauni

A cewar takardar, sanannun sakamakon da ke biyo bayan abc suna biyo baya daga sakamakonta ma, “da ƙayyadaddun adadi masu rauni”. Takardar ta tabbatar da misalai biyu:

• wani nau’i mai rauni na hasashen Szpiro (Szpiro’s conjecture) a kan lanƙwasai na elliptic (elliptic curves), da ninki 144 (1 + log 24) + ε;
• rashin wanzuwar abin da ake kira sifilolin Siegel (Siegel zeros) ga wani iyalin ayyuka (functions) na ka’idar lambobi, wato L-functions na quadratic characters masu discriminant mai alamar ragi (negative) — ta hanyar wata hujja ta Granville da Stark, wadda, kamar yadda takardar ta lura, ke aiki da ma’aunin ƙarfi da ba 1 + ε ba. Takardar kuma tana ambaton wasu preprint guda biyu na kwanan nan a kan wannan tambaya.

Har yanzu ana buƙatar dubawa

Wannan hujja ce mai shafi 39 ta mai rubutu guda, an rubuta ta da harshe mai zurfi na geometry na lissafin lambobi (arithmetic geometry), kuma ba ta riga ta bi ta hannun takwarorin masana (peer review) ba. Takardar tana godiya ga abokan aiki 26 saboda taimakonsu, amma hakan ba ya nufin sun duba ta ba. Takardar ba ta faɗi ko za a iya lissafa ƙayyadadden adadin A(ε) a fili (explicitly) ba.

Idan hujjar ta tabbata, jimlolin a + b = c za su sami rufi mai tsayayyen ma’aunin ƙarfi: ba wanda hasashen ke nema ba tukuna, amma 24.