WASANIN MONDRIAN: GIƁIN DA YA DAINA GIRMA

Mai zane ɗan ƙasar Holland Piet Mondrian ya fi shahara da ayyukansa na De Stijl, waɗanda galibi aka yi su da rectangle masu launi. Masana lissafi sun aro su don wani wasanin hankali da ke kama da mai sauƙi. Ɗauki zanen murabba’i mai gefen n. Yanka shi zuwa rectangle masu gefuna na lambobi cikakku, dukansu a jere da gefunan zanen. Doka ɗaya: babu rectangle biyu da za su yi siffa ɗaya, ko da bayan juya ɗaya kwata — 2 × 3 da 3 × 2 ba za su iya kasancewa tare a zane ɗaya ba.

Yanzu a kwatanta filayensu. Bambancin da ke tsakanin mafi girma da mafi ƙanƙanta shi ne naƙasu (defect). Tambayar ita ce yaya ƙanƙantarsa zai iya zama.

Yanka cikakke da babu wanda ya same shi

Yanka mai naƙasu sifili zai zama cikakke: rectangle dukansu sun bambanta a siffa, amma dukansu suna da fili ɗaya daidai. Daidaiton filaye kaɗai bai isa ba; dole ne guntayen su dace da juna ba tare da giɓi ko rufewa juna ba. Babu wanda ya sani ko akwai yanka cikakke ga kowane murabba’i. An kawar da shi ga kowane gefe har zuwa 1,000, kuma ra’ayin gaba ɗaya shi ne babu shi.

A bayan wannan ra’ayi akwai wani mafi faɗi: cewa mafi ƙanƙantar naƙasu mai yiwuwa, da ake rubutawa M(n), yana girma tare da girman murabba’in. Misalan da aka sani, har zuwa n = 900, sun yi kamar suna nuna haka. A wata takarda kafin bugawa (preprint) ta Satumba 2026, wata ƙungiya ta ba da rahoton iyakar sama ta farko mara sauƙi, M(n) ≤ 10 n^(5/6) — giɓin da har yanzu zai iya girma ba iyaka.

Rectangle bakwai da √19

Cooper O’Kuhn da Freeman Slaughter na Jami’ar South Florida sun fara daga wani abin ban sha’awa da aka sani: hanyar yanka murabba’i zuwa rectangle bakwai daban-daban masu fili ɗaya daidai. Matsalarsa ita ce gefunan ba lambobi cikakku ba ne; suna ƙunshe da √19, lamba mara hankali (irrational).

Ra’ayinsu shi ne a maye gurbin √19 da ɓangarorin p/q da ke ƙara kusantarsa. Kayan aikin da ya dace wata lissafi ce sananniya ta ka’idar lambobi, lissafin Pell (Pell’s equation): a nan, p² − 19q² dole ne ya tsaya a ƙayyadadden ƙima. Da wannan zaɓi, gefunan sun zama lambobi cikakku, kuma yaɗuwar filaye ya dogara ne kawai kan wannan ƙayyadadden ƙima — ba kan yadda lambobin suke girma ba. Sun nuna cewa naƙasun daidai yake da 140 × |p² − 19q²|.

Murabba’in 315

Mafi ƙanƙantar misali yana amfani da p = 13 da q = 3, inda 13² − 19 × 3² = −2. Yana ba da murabba’i mai gefen 315 da aka yanka zuwa:

259 × 55, 56 × 255, 238 × 60, 77 × 185, 189 × 75, 70 × 200 da 112 × 125.

Mafi girman fili shi ne 238 × 60 = 14,280; mafi ƙanƙanta, 70 × 200 = 14,000. Naƙasun shi ne 280. Filaye bakwai ɗin sun haɗu zuwa 99,225, wanda hakika shi ne 315 × 315.

280 har abada

Sannan ginin yana maimaituwa. Tsawon gefunan suna bin wata doka mai sauƙi: 315, sannan 106,785, sannan kowane sabon gefe ninki 340 ne na na baya debe wanda ke kafin shi — 36,306,585, sannan 12,344,132,115, haka nan ba ƙarshe. Kowane ɗaya daga cikin waɗannan murabba’ai ana iya yanka shi zuwa rectangle bakwai daban-daban da naƙasu daidai 280.

Don haka mafi ƙanƙantar naƙasu ba ya ci gaba da girma. A kalmomin marubutan, ba ma za a iya samun wani aiki (function) da ke nufin rashin iyaka — na layi, na logarithm ko wani abu daban — da ke zama ƙasa da M(n) ga kowane murabba’i mai isasshen girma ba. Wannan ya rushe wani hasashe (conjecture) da suka bayyana a matsayin a buɗe tun lokacin da aka fara gabatar da matsalar. Wani bambancin wannan hanya yana ba da, ga kowace lamba cikakkiya k, murabba’ai marasa iyaka waɗanda gefensu yake rabuwa da k, da naƙasu na 560 a mafi yawa.

Har yanzu babu murabba’i cikakke

Sakamakon ba ya faɗin komai game da murabba’ai da ke wajen wannan iyali, kuma ba ya kawar da yanka cikakke. Marubutan ma sun tabbatar da cewa irin wannan yanka na rectangle bakwai ba zai taɓa sauka ƙasa da 280 ba: naƙasu na sifili dole ne ya fito daga wani ra’ayi daban gaba ɗaya. Suna hasashen cewa 280 shi ne mafi kyawun abin da waɗannan rectangle bakwai za su iya yi, suna ba da shawarar cewa zane-zane masu siffar rectangle za su iya yin fiye da haka, kuma suna tsammanin “guguwar ci gaba mai sauri”. Ko akwai murabba’in Mondrian cikakke har yanzu a buɗe yake — amma hujjar cewa ba zai yiwu ba, saboda giɓin dole ne ya girma, ta tafi.