FENTA KOWACE LAMBA DA JA KO SHUƊI: SAKAMAKON NINKAWA MASU LAUNI ƊAYA NA IYA KUSAN ƁACEWA

Ɗauki alƙaluman zane (crayons) guda biyu, ɗaya ja ɗaya shuɗi, sannan fenta kowace cikakkiyar lamba mai kyau (positive whole number) da ɗaya daga cikinsu. Yanzu, kira lamba sakamakon ninkawa mai launi ɗaya idan za a iya rubuta ta a matsayin sakamakon ninka lambobi biyu ja ko lambobi biyu shuɗi — an yarda a yi amfani da lamba ɗaya sau biyu, don haka 4 = 2 × 2 kullum yana ƙidayuwa.

Uku cikin huɗu

Wani misali mai sauƙi ya nuna yadda abin yake aiki. Fenta lambobin mara da ja, lambobin cika kuma da shuɗi. Har zuwa 9, sakamakon ninkawa masu launi ɗaya su ne 1, 3, 5, 7 da 9 (mara sau mara), da 4 = 2 × 2 da 8 = 2 × 4 (cika sau cika): bakwai cikin tara. Lambar 2 ta rasa shiga, saboda rarrabawarta (factorisation) kaɗai, 1 × 2, tana haɗa launuka; haka ma 6 = 1 × 6 = 2 × 3. Gaba ɗaya, yawansu har zuwa n shi ne cikakken ɓangaren (whole part) (3n + 1)/4, don haka kason yana kusantar kashi uku cikin huɗu. Waɗanda suka rasa su ne 2, 6, 10, 14… — lamba ɗaya cikin huɗu.

Shin tsarin launi mafi wayo zai iya yin abin da ya fi wannan nesa? An riga an san cewa ga kowane tsarin launi tsayayye (fixed), sakamakon ninkawa masu launi ɗaya kullum suna zama wani kaso mai kyau (positive share) na lambobin. Ainihin tambayar, wadda Erdős da Sárközy suka gabatar a 1990 kuma suka sake gabatarwa a 1993, kuma take cikin shahararren littafin Richard Guy na matsalolin da ba a warware ba a ka’idar lambobi (number theory), ita ce ko akwai ƙasan bai ɗaya (universal floor): wani tsayayyen kaso da dole ne kowane tsarin launi ya kai. A faɗe shi ta wata fuska: ga kowace ƙaramar lamba ε, shin akwai tsarin launi wanda kason sakamakon ninkawa masu launi ɗaya a cikinsa, aƙalla a wasu sassa, ke sauka ƙasa da ε? Matsalar tana cikin ka’idar Ramsey ta ninkawa (multiplicative Ramsey theory), wadda ke nazarin tsare-tsaren da babu tsarin launi da zai iya gujewa.

Babu ƙasa

Takardar John M. Campbell, a Dalhousie University da ke Halifax, Kanada, ta tabbatar da cewa babu ƙasan bai ɗaya. Ga kowane ε, komai ƙanƙantarsa, akwai tsarin launin ja da shuɗi na cikakkun lambobi wanda, a kan jerin wuraren yankewa (cut-off points) marasa iyaka, ƙasa da juzu’in ε na lambobin har zuwa wannan wuri ne sakamakon ninkawa masu launi ɗaya.

Babban ra’ayin shi ne madubi. Ginin yana zaɓar jerin wuraren yankewa masu ƙaruwa N₁, N₂, N₃… da kuma wasu wuraren sauyi inda launin ke juyawa. Ana zaɓar su ta yadda, ƙasa da kowane wurin yankewa N, lamba x da madubinta N/x kullum suna da launuka masu saɓani. Sakamakon ninkawa a × b da ke kusa da N yana haɗa ƙaramar lamba da babba wadda kusan madubinta ce, don haka abubuwa biyun da aka ninka (factors) kusan kullum suna da launuka daban-daban. Abubuwan da ba kasafai ake samu ba da suka saɓa — lambobin da ke da mai raba su (divisor) a cikin wani ƙunƙuntaccen kewayo — ana ƙidaya su da kayan aikin ka’idar lambobi ta nazari (analytic number theory), ciki har da gaskiyar, da ake danganta wa Euler, cewa jimlar juye-juyen (reciprocals) lambobin firamare (primes), 1/2 + 1/3 + 1/5 + 1/7 + …, tana girma ba tare da iyaka ba.

Tsarin launi da ba za a iya kaiwa ba

Tsarin launin yana nan, amma babu wanda zai taɓa ganinsa yana aiki. Dole ne a zaɓi wuraren yankewarsa masu girma ƙwarai, kuma takardar ta lissafa girmansu ga burin ε = 1/10. Dole ne wurin yankewa na farko ya wuce wata lamba da ke tsakanin exp(exp(96,303.3504)) da exp(exp(96,303.3505)). Sakamakon haka, idan an fenta 1 da ja, lambobi 10^(10^41,823) na farko duka ja ne — yawan da shi kansa yake da lambobi (digits) fiye da 10^41,823. Sai can nesa da wannan ne dabarar madubin ke fara yin tasiri. Babu kwamfutar da za ta taɓa duba irin wannan tsarin launi ta hanyar ƙidaya.

Amsa da aka samo tare da AI

Takardar ba ta ɓoye yadda amsar ta zo ba. A ranar 18 ga Oktoba 2026, matsalar daga littafin Guy ta isa ga wani samfurin AI, GPT-6 Astra Pro, daga hannun Campbell, tare da buƙatar ya warware ta. Bayan minti 13 da daƙiƙa 53 na “tunani”, samfurin ya amsa cewa babu irin wannan constant (tsayayyiyar lamba) kuma ya gabatar da wani gini. Ta hanyar dogon musayar ra’ayi, gyare-gyare da bincike, daga nan aka rubuta hujjar, kuma an danganta wasu daga cikin matakanta — wani nau’in Markov’s inequality mara dogaro da yiwuwa (non-probabilistic), nazarin al’amura ɗaya bayan ɗaya (case analysis), wani rashin daidaito (inequality) a kan aikin digamma (digamma function) — da binciken Campbell na kashin kai. A cewar Campbell, an sake duba kowace shawarar AI kuma an tantance ta daban, kuma Campbell ne ke ɗaukar cikakken alhakin lissafin.

Hujjar preprint ce ta mutum guda, kuma wasu masana lissafi ba su riga sun duba ta ba. Ta ƙare da sababbin tambayoyi: me zai faru idan lambobin firamare kaɗai aka fenta, kuma waɗanne tsare-tsaren launi ne ke sa kason sakamakon ninkawa masu launi ɗaya ya tsaya a kan wata ƙima da aka bayar — ko kuma kada ya taɓa tsayawa kwata-kwata?