LAMBOBI 537, LAUNUKA SHIDA, BABU JIMLA

Ga wani kacici-kacici da yaro zai iya fahimta. Ɗauki lambobi 1, 2, 3, da sauransu. Ba kowacce launi. Doka ɗaya kawai: a cikin launi ɗaya, ba za ka taɓa samun lambobi biyu tare da jimlarsu ba. Lambobin biyu ma za su iya zama iri ɗaya, don haka idan 2 ja ne, to 4 ba zai iya zama ja ba, saboda 2 + 2 = 4.

Tarin da ba shi da irin wannan uku-uku ana kiransa mara jimla (sum-free). Har ina za ka iya zuwa kafin doka ta tilasta maka tsayawa? Wannan iyaka, ga launuka k, ita ce lambar Schur S(k). Matsalar ta samo asali daga wata takarda ta 1916 ta I. Schur, abu na farko a cikin jerin manazarta.

Jeri sananne da taurin kai

Da launi ɗaya, kana makalewa nan take: 1 + 1 = 2, don haka launin zai iya ɗaukar lamba 1 kawai. Ƙimomin da aka sani su ne:

S(1) = 1
S(2) = 4
S(3) = 13
S(4) = 44
S(5) = 160, wanda M. J. H. Heule ya tantance (a cikin takardar da aka wallafa a 2018).

Ga launuka shida, babu wanda ya san amsar. Tun 2000, mafi kyawun gini da aka sani, wanda H. Fredricksen da M. M. Sweet suka samo ta hanyar bincike a kan launuka masu daidaito (symmetric), ya kai 536. Tun daga lokacin ake ambaton wannan lamba a matsayin mafi kyawun iyakar ƙasa — wato bene: S(6) ya kai aƙalla wannan girma, kuma ainihin ƙimarsa, wadda har yanzu ba a sani ba, na iya fin haka.

Ƙarin lamba ɗaya

Zhiyang Chen da Fan Long, na Jami’ar Toronto, da Ligeng Zhu, na NVIDIA Research, yanzu sun tura ta zuwa 537. Hujjarsu irin wadda masana lissafi ke so ce saboda rashin kewayarta: sun buga amsar. Jerin lambobi shida — masu lambobi 76, 76, 88, 70, 76 da 151 — da tare suke ƙunshe da kowace lamba cikakka daga 1 zuwa 537 sau ɗaya daidai, ba tare da wani launi da ke ɗauke da x, y da x + y ba. Duba ta yana nufin gwada kowace jimla a cikin kowane jeri, aiki mai sauri ga kwamfuta.

Mafitar tana da siffa mai ban mamaki. Tana da daidaito (symmetric): kowace lamba x tana da launi ɗaya da 538 − x. Launuka huɗu na farko suna farawa a 1, 2, 3 da 4; na biyar yana bayyana ne kawai daga 38, na shida kawai daga 151, kuma launi na shida yana cika dukan tsakiyar daga 237 zuwa 301. Har launi na farko ana iya sake gina shi daga iri na ƙananan lambobi 14 kawai ta hanyar juyawar madubi, sauyin wuri (shift) da ƙarin lamba ɗaya.

Raƙuman da ke yaɗuwa sama

Lambobin Schur suna ciyar da juna. Wasu dabarun da aka sani suna mayar da kyakkyawan launi na launuka k zuwa na launuka k + 2 ko k + 5: S(k + 2) aƙalla 10 S(k) + 2 ne, kuma S(k + 5) aƙalla 380 S(k) + 148. Don haka sabon 537 yana ɗaga wasu tarihi:

launuka takwas: daga 5,362 zuwa 5,372 — kuma ƙungiyar ta gina wannan launi a fili ita ma;
launuka goma sha ɗaya: daga 203,828 zuwa 204,208;
launuka goma sha uku: daga 2,038,282 zuwa 2,042,082.

Hakanan yana inganta wata lambar Ramsey, R₆(3): mafi ƙarancin adadin maki, waɗanda aka haɗa dukansu biyu-biyu, ta yadda duk yadda aka ba haɗin launi da launuka shida, dole a sami triangle na launi ɗaya. Ka sa wa maki 538 lamba daga 0 zuwa 537, kuma ka ba haɗin tsakanin i da j launin bambancinsu |i − j|: launi mara jimla ba ya barin ko triangle ɗaya na launi ɗaya. Don haka R₆(3) ya kai aƙalla 539, maimakon 538 a da.

Hanyoyi 131 na yin sa

Binciken bai sami mafita ɗaya ba sai 131 da suka bambanta da gaske, waɗanda marubutan suka rarraba zuwa iyalai biyar kuma suka wallafa gaba ɗaya. Yawancinsu suna kusa sosai da ta farko: waɗanda ke cikin iyalan F1 zuwa F4 sun yarda da ita a aƙalla wurare 462 daga cikin 537. Iyali ɗaya yana karya daidaiton madubi; wani kuma an gina shi ta hanyar haɗa kwafi uku da aka gauraya na asalin.

Takardar ba ta bayyana yadda aka gudanar da binciken kansa ba. Marubutan sun jaddada cewa launukansu 131 ba cikakken kundin mafita na launuka shida ba ne, kawai sabbin wuraren farawa ne ga duk wanda ke son ƙoƙarin kaiwa 538.