Beautiful Number PatternsYue Kwok Choy

1. Towers

1

1 x 8 + 1 = 912 x 8 + 2 = 98

123 x 8 + 3 = 9871234 x 8 + 4 = 9876

12345 x 8 + 5 = 98765123456 x 8 + 6 = 987654

1234567 x 8 + 7 = 987654312345678 x 8 + 8 = 98765432

123456789 x 8 + 9 = 987654321

2. Product

3. Need your algebra proofs

2

1 + 2 = 34 + 5 + 6 = 7 + 8

9 + 10 + 11 + 12 = 13 + 14 + 1516 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24

….(Can you prove it?)

492+502=49014992+5002=499001

49992+50002=49990001499992+500002=4999900001

….

991 × 111 = 110001991 × 1111 = 1101001

991 × 11111 = 11011001991 × 111111 = 110111001

991 × 1111111 = 1101111001….

32+2=11332+22=1111

3332+222=11111133332+2222=11111111

….

12=122=1+3=1+2+1

32=1+3+5=1+2+3+2+142=1+3+5+7=1+2+3+4+3+2+1

52=1+3+5+7+9=1+2+3+4+5+4+3+2+162=1+3+5+7+9+11=1+2+3+4+5+6+5+4+3+2+1

….

4. Powers

5. Factorials

6. Cyclic

3

0588235294117647 x 1 = 0588235294117647 0588235294117647 x 2 = 1176470588235294 0588235294117647 x 3 = 1764705882352941 0588235294117647 x 4 = 2352941176470588 0588235294117647 x 5 = 2941176470588235 0588235294117647 x 6 = 3529411764705882

12+56+64=24+32+76122+562+642=242+322+762

123+561+642=242+323+7611232+5612+6422=2422+3232+7612

1237+5619+6428=2428+3237+761912372+56192+64282=24282+32372+76192

1+6+7+17+18+23=2+3+11+13+21+2212+62+72+112+182+232=22+32+112+132+212++222

13+63+73+113+183+233=23+33+113+133+213++223

14+64+74+114+184+234=24+34+114+134+214++224

15+65+75+115+185+235=25+35+115+135+215++225

1676=11+62+73+64=15+64+73+62

2646798=21+62+43+64+75+96+87

444=(41+41+41 )+(42+42+42 )+(43+43+43 )+(44+44+44 )

0! × 1! = 1!1! × 2! = 2!

6! × 7! = 10!1! × 3! × 5! = 6!

1! × 3! × 5! × 7! = 10!

1=1!2=2!

145=1!+4!+5!40,585=4!+0!+5!+8!+5!

8. Prime

4

142857 × 1 ＝142857 142857 × 2＝285714 142857 × 3＝428571 142857 × 4＝571428 142857 × 5＝714285 142857 × 6＝857142

0588235294117647 x 1 = 0588235294117647 0588235294117647 x 2 = 1176470588235294 0588235294117647 x 3 = 1764705882352941 0588235294117647 x 4 = 2352941176470588 0588235294117647 x 5 = 2941176470588235 0588235294117647 x 6 = 3529411764705882

If A = 1, B = 2, C = 3, …, Z = 26, then PRIME =16 + 18 + 9 + 13 + 5 = 61 which is prime!

Smallest prime of digits 1 to 152

11101

100910007

1000031000003

10000019100000007

100000000710000000019

1000000000031000000000039

10000000000037100000000000031

Largestest prime of digits 1 to 15797997997399991999983999999199999989999999937999999996799999999977999999999989999999999997199999999999973999999999999989

Smallest Palindromic Prime Tower2

11101

103011003001

Largest Palindromic Prime Tower2

11929

9868999898999

9. Some More

5

Smallest Palindromic Prime Tower2

11101

103011003001

Largest Palindromic Prime Tower2

11929

9868999898999

They are prime numbers123456789112345678901234567891123456789123456789123456789128116440335967=214+814+114+114+614+414+414+014+314+314+514+914+614+714 Where the length of digits = 14

2187 + 1234= 34212187 + 12345 = 14532

2187 + 123456 = 1256432187 + 1234567 = 1236754

2187 + 12345678 = 123478652187 + 123456789 = 123458976

2=134586729 , 3=

174695823 , 4=

157683942

5=134582697 , 6=

176582943 , 7=

167582394 , 8=

254963187

9=574296381

=9752410836

=9852310647

=9574210638

=752948361

=582396471

10. Internet search and learn more for interest

6

Proper divisorsPerfect numbersFriendly numbersAmicable numbers

Prime numbersTwin prime numbersMersenne prime numbersFactorial prime numbersLargest prime number known

Large numbersBillion, TrillionGoogolGoogolplexSkewes' number Moser's numberGraham's number

Interesting numbers and constantsPi, Phi, , Golden RatioeSquare root of 2, √2Complex number, iFibonacci numbersHardy–Ramanujan numberTaxicab numberKaprekar's constant and “black holes” of numbers