0:00:04.420,0:00:07.221
Tasavvur qiling, milodan avvalgi yillar.

0:00:07.221,0:00:09.468
Endi o'ylab ko'ring:

0:00:09.468,0:00:12.721
Qanday qilib soatsiz vaqtni aniqlashgan?

0:00:12.721,0:00:15.315
Barcha soatlar vaqt oqimini [br]teng bo'laklarga bo'luvchi

0:00:15.315,0:00:18.890
qandaydir takroriy shaklga asoslangan.

0:00:18.890,0:00:20.688
Bunday takroriy shakillarni topish uchun

0:00:20.688,0:00:22.918
samolarga yuzlanamiz.

0:00:22.918,0:00:24.902
Har kuni, quyoshning chiqishi va botishi

0:00:24.902,0:00:26.184
bunday shakllarning eng oddiysidir.

0:00:26.184,0:00:28.760
Lekin uzoqroq vaqt bo'lagini kuzatish uchun

0:00:28.760,0:00:30.811
uzoqroq takrorlanishlarga e'tibor beramiz.

0:00:30.811,0:00:32.512
Buning uchun esa oyga yuzlanamiz.

0:00:32.512,0:00:33.853
E'tibor bergan bo'lsangiz,

0:00:33.853,0:00:36.578
oy kunlar osha to'lishadi va kichrayadi.

0:00:36.578,0:00:37.894
To'lin oylar orasidagi kunlar sonini

0:00:37.894,0:00:38.978
sanaydigan bo'lsak,

0:00:38.978,0:00:40.910
u 29 kunga teng.

0:00:40.910,0:00:42.833
Bir oydagi kunlar soni shundan [br]kelib chiqqan bo'lsa kerak.

0:00:42.833,0:00:45.873
Ammo 29 ni teng bo'laklarga [br]bo'lishga harakat qilsak,

0:00:45.873,0:00:49.227
bir muammoga duch kelamiz: [br]buning iloji yo'q.

0:00:49.227,0:00:51.676
29 ni teng bo'laklarga bo'lishning [br]yagona yo'li

0:00:51.676,0:00:54.819
uni 29 ta teng bo'lakka bo'lishdan iborat.

0:00:54.819,0:00:57.102
29 soni tub son hisoblanadi.

0:00:57.102,0:00:59.061
Uni bo'linmas deb tasavvur qiling.

0:00:59.061,0:01:00.879
Agar son birdan boshqa

0:01:00.879,0:01:02.814
teng bo'laklarga bo'linsa,

0:01:02.814,0:01:04.621
biz uni 'murakkab son' deb ataymiz.

0:01:04.621,0:01:06.608
Endi biz qiziqishimiz mumkin,

0:01:06.608,0:01:08.450
"Dunyoda nechta tub son bo'lishi mumkin?

0:01:08.450,0:01:10.398
Va ularning eng kattasi [br]nechaga teng ekan?"

0:01:10.398,0:01:13.744
Keling, barcha sonlarni [br]ikkita guruhga bo'lamiz.

0:01:13.744,0:01:15.611
Tub sonlar chap tomonda

0:01:15.611,0:01:17.648
va murakkab sonlar o'ng tomonda.

0:01:17.648,0:01:20.379
Boshida, u tomondan bu tomonga raqs [br]tushayotganga o'xshaydilar.

0:01:20.379,0:01:23.017
Ammo ularning joylashuvida [br]aniq bir shakl mavjud emas.

0:01:23.017,0:01:24.439
Keling, bunday shaklni ko'rish uchun

0:01:24.439,0:01:26.077
zamonaviy usuldan foydalanamiz.

0:01:26.077,0:01:29.047
Bu usul "Ulam spirali" deb nomlanadi.

0:01:29.047,0:01:32.011
Boshida, barcha raqamlarni tartib bilan[br]o'sayotgan spiral

0:01:32.011,0:01:34.043
ichiga joylab chiqamiz.

0:01:34.043,0:01:37.164
Keyin, barcha tub sonlarni [br]ko'k ranga bo'yab chiqamiz.

0:01:37.164,0:01:41.290
Nihoyat, biz millionlab raqamlarni[br]ko'rish uchun uzoqlashamiz.

0:01:41.290,0:01:42.860
Mana bu tugalmas tub sonlarning

0:01:42.860,0:01:45.365
shakli hisoblanadi.

0:01:45.365,0:01:47.967
Hayratlanarlisi shuki, bu shaklning [br]tuliq strukturasi

0:01:47.967,0:01:50.314
haligacha topilmagan.

0:01:50.314,0:01:51.843
Nimanidir kashf etish arafasida[br]turganga o'xshaymiz...

0:01:51.843,0:01:52.987
Keling, m.a. 300 yillarga,

0:01:52.987,0:01:55.526
Qadimgi Gretsiyaga sayr qilamiz.

0:01:55.526,0:01:58.183
Buyuk faylasuf, Aleksandryalik Evklid,

0:01:58.183,0:01:59.411
barcha sonlarni

0:01:59.411,0:02:02.607
bu ikki guruhga ajralishini anglab yetadi.

0:02:02.607,0:02:04.896
Dastlab, u istalgan sonni

0:02:04.896,0:02:07.078
kichik bo'linmas teng [br]sonlar guruhlarigacha

0:02:07.078,0:02:10.599
bo'lish mumkinligini anglab yetadi.

0:02:10.599,0:02:12.921
Va bu eng kichik sonlar esa, har doim

0:02:12.921,0:02:15.760
tub sonlardir.

0:02:15.760,0:02:17.148
Shunday qilib, u barcha sonlar

0:02:17.148,0:02:20.542
tub sonlardan qurilganini tushunib yetadi.

0:02:20.542,0:02:23.317
Aniqrog'i, barcha sonlar olamini [br]tasavvur qiling,

0:02:23.317,0:02:25.674
tub sonlar haqida unuting.

0:02:25.674,0:02:28.037
Endi istalgan murakkab sonni olamiz

0:02:28.037,0:02:30.518
va bo'laklarga ajratamiz

0:02:30.518,0:02:33.354
va bu bo'laklar, har doim [br]tub sonlardir.

0:02:33.354,0:02:34.774
Demak, Evklid istalgan raqam

0:02:34.774,0:02:37.415
kichikroq tub sonlar guruhi orqali ifodalanishi [br]mumkinligin tushunib yetgan.

0:02:37.675,0:02:39.145
Ularni g'ishtlar deb tasavvur qiling.

0:02:40.221,0:02:41.969
Qaysi son bo'lishidan qat'iy nazar,

0:02:41.999,0:02:43.937
uni kichiroq tub sonlarni qo'shish [br]bilan yasash mumkin.

0:02:46.157,0:02:47.972
Mana shu Evklid kashfiyotining [br]asosi bo'lib,

0:02:48.042,0:02:50.759
"Arifmetikaning asosiy nazariyasi" [br]deb nomlanadi.

0:02:50.759,0:02:52.013
Unga ko'ra,

0:02:52.013,0:02:53.934
istalgan raqamni, aytaylik, 30 ni olamiz

0:02:53.934,0:02:55.501
va uning tub ko'paytuvchilarini topamiz.

0:02:55.501,0:02:57.233
30 teng bo'linadi.

0:02:57.233,0:02:59.763
Buni biz "ko'paytuvchilarga ajratish" [br]deb ataymiz.

0:02:59.763,0:03:01.624
Bu bizga tub ko'paytuvchilarni [br]topish imkonini beradi.

0:03:01.624,0:03:05.811
Bizning holatda 2,3 va 5[br]30 ning tub kupaytuvchilaridir.

0:03:05.811,0:03:07.906
Evklid yana shuni tushunib yetdiki, [br]sonning tub ko'paytuvchilarini

0:03:07.906,0:03:10.714
bir necha bor ko'paytirish orqali

0:03:10.714,0:03:12.739
dastlabki sonni keltirib chiqarish [br]mumkin ekan.

0:03:12.739,0:03:13.780
30 sonini yasash uchun esa [br]uning tub ko'paytuvchilarini

0:03:13.780,0:03:16.178
bir martadan ko'paytirish kifoya.

0:03:16.178,0:03:20.158
2 x 3 x 5[br]30 soning tub kupaytuvchilaridir.

0:03:20.158,0:03:23.153
Bularni o'ziga hos kalit yoki [br]kombinatsiya deyish mumkin.

0:03:23.153,0:03:24.887
30 sonini boshqa tub son guruhlari

0:03:24.887,0:03:27.110
ko'paytmasi orqali yasashning

0:03:27.110,0:03:28.792
imkoni yo'q.

0:03:28.792,0:03:31.276
Shunday qilib, istalgan son [br]faqat va faqat

0:03:31.276,0:03:34.046
bitta yo'l bilan [br]tub ko'paytuvchilarga ajraladi.

0:03:34.046,0:03:36.299
Misol uchun, har bir sonni

0:03:36.299,0:03:38.017
alohida qulf deb tasavvur qiling.

0:03:38.033,0:03:39.722
Har bir qulfning (sonning) kaliti

0:03:39.722,0:03:41.882
uning tub ko'paytuvchilari bo'ladi.

0:03:41.940,0:03:43.897
Hech bir qulf bir xil kalitga ega emas.

0:03:43.937,0:03:47.937
Hech bir son bir xil tub ko'paytuvchilardan [br]tashkil topmaydi.