00 Mathematics by Adi Cox 00

8/11/2019 00 Mathematics by Adi Cox 00

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Mathematics By Adi Cox

CONTENT

1-Encrypt2-The-4-x-4-Magic-Square3-The-Hour-And-Minute-Hand-Time-Problem4-An-Introduction-to-Stifled-Numbers5-S-1-Stifled-Minus-One-Numbers6-Finding-Fractions-in-Other-Bases(1)7-Chaos8-Factorizing-Cubic-Equations-of-the-Form-Ax-3-Bx-2-Cx-D-09-Matrices(1)10-Problem-2-500-Factors11-Problem-1-1st-May-201312-The-Investigation-Of-A-Tetrahedron-By-Adi-Cox13-The-4-x-4-Magic-Square-214-Noughts-and-Crosses-Orbits-and-Stabilizers(1)15-The-Family-of-Adi-Polytopes(1)16-Find-The-Simplex-Edge-Length-X17-Applications-and-Non-Euclidean-Geometry18-Chromogeometry


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Encryption

The first 26 prime numbers represent thealphabet respectively.

A = 2

B = 3C = 5D = 7E = 11F = 13G = 17H = 19I = 23J = 29K = 31L = 37M = 41

N = 43O = 47P = 53Q = 59R = 61S = 67T = 71U = 73V = 79W = 83X = 89Y = 97Z = 101

Find the prime factors of the numbersbelow to get the anagrams of 15 wordswhich make a poem.

292691086560312749111557111616706109277781342141862089191961127274312321795608651041445253063762414463279070946


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The 4 x 4 Magic Square by Adi Cox----------------------------------------------------------------

To find the value that a line equals in a 4x4 magic square withnumbers from 1 to 16 we add those numbers from 1 to 16 thendivide the answer by 4.

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16=136136/4=34

_________________________________| | | | || 4 | 14 | 15 | 1 ||_______|_______|_______|_______|| | | | || 9 | 7 | 6 | 12 ||_______|_______|_______|_______|| | | | || 5 | 11 | 10 | 8 ||_______|_______|_______|_______|| | | | || 16 | 2 | 3 | 13 ||_______|_______|_______|_______|

The html file that finds the 4 partition that adds up to 34:

----------------------------------------------------------------

A for loop

document.write("An Investigation Of 4 Partitions")

for (a = 1; a


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}

}}

----------------------------------------------------------------

The above html file gives the following 86 results:

1 + 2 + 15 + 16 = 341 + 3 + 14 + 16 = 341 + 4 + 13 + 16 = 341 + 4 + 14 + 15 = 341 + 5 + 12 + 16 = 341 + 5 + 13 + 15 = 341 + 6 + 11 + 16 = 341 + 6 + 12 + 15 = 341 + 6 + 13 + 14 = 341 + 7 + 10 + 16 = 34

1 + 7 + 11 + 15 = 341 + 7 + 12 + 14 = 341 + 8 + 9 + 16 = 341 + 8 + 10 + 15 = 341 + 8 + 11 + 14 = 341 + 8 + 12 + 13 = 341 + 9 + 10 + 14 = 341 + 9 + 11 + 13 = 341 + 10 + 11 + 12 = 342 + 3 + 13 + 16 = 342 + 3 + 14 + 15 = 342 + 4 + 12 + 16 = 342 + 4 + 13 + 15 = 34

2 + 5 + 11 + 16 = 342 + 5 + 12 + 15 = 342 + 5 + 13 + 14 = 342 + 6 + 10 + 16 = 342 + 6 + 11 + 15 = 342 + 6 + 12 + 14 = 342 + 7 + 9 + 16 = 342 + 7 + 10 + 15 = 342 + 7 + 11 + 14 = 342 + 7 + 12 + 13 = 342 + 8 + 9 + 15 = 342 + 8 + 10 + 14 = 342 + 8 + 11 + 13 = 34

2 + 9 + 10 + 13 = 342 + 9 + 11 + 12 = 343 + 4 + 11 + 16 = 343 + 4 + 12 + 15 = 343 + 4 + 13 + 14 = 343 + 5 + 10 + 16 = 343 + 5 + 11 + 15 = 343 + 5 + 12 + 14 = 343 + 6 + 9 + 16 = 343 + 6 + 10 + 15 = 34


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3 + 6 + 11 + 14 = 343 + 6 + 12 + 13 = 343 + 7 + 8 + 16 = 343 + 7 + 9 + 15 = 343 + 7 + 10 + 14 = 343 + 7 + 11 + 13 = 343 + 8 + 9 + 14 = 343 + 8 + 10 + 13 = 343 + 8 + 11 + 12 = 343 + 9 + 10 + 12 = 344 + 5 + 9 + 16 = 344 + 5 + 10 + 15 = 344 + 5 + 11 + 14 = 344 + 5 + 12 + 13 = 344 + 6 + 8 + 16 = 344 + 6 + 9 + 15 = 344 + 6 + 10 + 14 = 344 + 6 + 11 + 13 = 344 + 7 + 8 + 15 = 344 + 7 + 9 + 14 = 344 + 7 + 10 + 13 = 344 + 7 + 11 + 12 = 344 + 8 + 9 + 13 = 344 + 8 + 10 + 12 = 34

4 + 9 + 10 + 11 = 345 + 6 + 7 + 16 = 345 + 6 + 8 + 15 = 345 + 6 + 9 + 14 = 345 + 6 + 10 + 13 = 345 + 6 + 11 + 12 = 345 + 7 + 8 + 14 = 345 + 7 + 9 + 13 = 345 + 7 + 10 + 12 = 345 + 8 + 9 + 12 = 345 + 8 + 10 + 11 = 346 + 7 + 8 + 13 = 346 + 7 + 9 + 12 = 34

6 + 7 + 10 + 11 = 346 + 8 + 9 + 11 = 347 + 8 + 9 + 10 = 34

----------------------------------------------------------------

The patterns that the 4 partitions above make on the magicsquare below, in their respective order.

_________________________________| | | | || 4 | 14 | 15 | 1 ||_______|_______|_______|_______|

| | | | || 9 | 7 | 6 | 12 ||_______|_______|_______|_______|| | | | || 5 | 11 | 10 | 8 ||_______|_______|_______|_______|| | | | || 16 | 2 | 3 | 13 ||_______|_______|_______|_______|


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O O O OO O O O

(13) O O O XX O O O 1 + 8 + 9 + 16 = 34O O O XX O O O

(14) O O X XO O O O 1 + 8 + 10 + 15 = 34O O X XO O O O

(15) O X O XO O O O 1 + 8 + 11 + 14 = 34O X O XO O O O

(16) O O O XO O O X 1 + 8 + 12 + 13 = 34O O O XO O O X

(17) O X O X

X O O O 1 + 9 + 10 + 14 = 34O O X OO O O O

(18) O O O XX O O O 1 + 9 + 11 + 13 = 34O X O OO O O X

(19) O O O XO O O X 1 + 10 + 11 + 12 = 34O X X OO O O O

(20) O O O OO O O O 2 + 3 + 13 + 16 = 34O O O OX X X X

(21) O X X OO O O O 2 + 3 + 14 + 15 = 34O O O OO X X O

(22) X O O OO O O X 2 + 4 + 12 + 16 = 34

O O O OX X O O

(23) X O X OO O O O 2 + 4 + 13 + 15 = 34O O O OO X O X

(24) O O O OO O O O 2 + 5 + 11 + 16 = 34


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X X O OX X O O

(25) O O X OO O O X 2 + 5 + 12 + 15 = 34X O O OO X O O

(26) O X O OO O O O 2 + 5 + 13 + 14 = 34X O O OO X O X

(27) O O O OO O X O 2 + 6 + 10 + 16 = 34O O X OX X O O

(28) O O X OO O X O 2 + 6 + 11 + 15 = 34O X O OO X O O

(29) O X O O

O O X X 2 + 6 + 12 + 14 = 34O O O OO X O O

(30) O O O OX X O O 2 + 7 + 9 + 16 = 34O O O OX X O O

(31) O O X OO X O O 2 + 7 + 10 + 15 = 34O O X OO X O O

(32) O X O OO X O O 2 + 7 + 11 + 14 = 34O X O OO X O O

(33) O O O OO X O X 2 + 7 + 12 + 13 = 34O O O OO X O X

(34) O O X OX O O O 2 + 8 + 9 + 15 = 34

O O O XO X O O

(35) O X O OO O O O 2 + 8 + 10 + 14 = 34O O X XO X O O

(36) O O O OO O O O 2 + 8 + 11 + 13 = 34


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O O O OO O X X

(49) O O O OO X O O 3 + 7 + 8 + 16 = 34O O O XX O X O

(50) O O X OX X O O 3 + 7 + 9 + 15 = 34O O O OO O X O

(51) O X O OO X O O 3 + 7 + 10 + 14 = 34O O X OO O X O

(52) O O O OO X O O 3 + 7 + 11 + 13 = 34O X O OO O X X

(53) O X O O

X O O O 3 + 8 + 9 + 14 = 34O O O XO O X O

(54) O O O OO O O O 3 + 8 + 10 + 13 = 34O O X XO O X X

(55) O O O OO O O X 3 + 8 + 11 + 12 = 34O X O XO O X O

(56) O O O OX O O X 3 + 9 + 10 + 12 = 34O O X OO O X O

(57) X O O OX O O O 4 + 5 + 9 + 16 = 34X O O OX O O O

(58) X O X OO O O O 4 + 5 + 10 + 15 = 34

X O X OO O O O

(59) X X O OO O O O 4 + 5 + 11 + 14 = 34X X O OO O O O

(60) X O O OO O O X 4 + 5 + 12 + 13 = 34


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X O O OO O O X

(61) X O O OO O X O 4 + 6 + 8 + 16 = 34O O O XX O O O

(62) X O X OX O X O 4 + 6 + 9 + 15 = 34O O O OO O O O

(63) X X O OO O X O 4 + 6 + 10 + 14 = 34O O X OO O O O

(64) X O O OO O X O 4 + 6 + 11 + 13 = 34O X O OO O O X

(65) X O X O

O X O O 4 + 7 + 8 + 15 = 34O O O XO O O O

(66) X X O OX X O O 4 + 7 + 9 + 14 = 34O O O OO O O O

(67) X O O OO X O O 4 + 7 + 10 + 13 = 34O O X OO O O X

(68) X O O OO X O X 4 + 7 + 11 + 12 = 34O X O OO O O O

(69) X O O OX O O O 4 + 8 + 9 + 13 = 34O O O XO O O X

(70) X O O OO O O X 4 + 8 + 10 + 12 = 34

O O X XO O O O

(71) X O O OX O O O 4 + 9 + 10 + 11 = 34O X X OO O O O

(72) O O O OO X X O 5 + 6 + 7 + 16 = 34


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X O O OX O O O

(73) O O X OO O X O 5 + 6 + 8 + 15 = 34X O O XO O O O

(74) O X O OX O X O 5 + 6 + 9 + 14 = 34X O O OO O O O

(75) O O O OO O X O 5 + 6 + 10 + 13 = 34X O X OO O O X

(76) O O O OO O X X 5 + 6 + 11 + 12 = 34X X O OO O O O

(77) O X O O

O X O O 5 + 7 + 8 + 14 = 34X O O XO O O O

(78) O O O OX X O O 5 + 7 + 9 + 13 = 34X O O OO O O X

(79) O O O OO X O X 5 + 7 + 10 + 12 = 34X O X OO O O O

(80) O O O OX O O X 5 + 8 + 9 + 12 = 34X O O XO O O O

(81) O O O OO O O O 5 + 8 + 10 + 11 = 34X X X XO O O O

(82) O O O OO X X O 6 + 7 + 8 + 13 = 34

O O O XO O O X

(83) O O O OX X X X 6 + 7 + 9 + 12 = 34O O O OO O O O

(84) O O O OO X X O 6 + 7 + 10 + 11 = 34


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O X X OO O O O

(85) O O O OX O X O 6 + 8 + 9 + 11 = 34O X O XO O O O

(86) O O O OX X O O 7 + 8 + 9 + 10 = 34O O X XO O O O


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PROBLEM:

At what times should the hour and minute hand overlapon a normal analogue clock, to the nearest second?

SOLUTION:

The number of seconds The time isafter twelve O'clock

120(360/11)1 = 3927.'27' 01hours 05min 27sec120(360/11)2 = 7854.'54' 02hours 10min 55sec120(360/11)3 = 11781.'81' 03hours 16min 28sec120(360/11)4 = 15709.'09' 04hours 21min 49sec120(360/11)5 = 19636.'36' 05hours 27min 16sec120(360/11)6 = 23563.'63' 06hours 32min 44sec120(360/11)7 = 27490.'90' 07hours 38min 11sec120(360/11)8 = 31418.'18' 08hours 43min 38sec120(360/11)9 = 35345.'45' 09hours 49min 05sec120(360/11)10 = 39272.'72' 10hours 54min 33sec120(360/11)11 = 43200 12hours 00min 00sec


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An Introduction To Stifled Numbers By Adi Cox 9/2/2013

______________________________________________________________________________

Stifled Numbers are numbers whose digits add up to its base. So there can be nosingle digit stifled number in any base. In base ten the set 2s10 has 9 twodigit stifled numbers:

2s10 = {19, 28, 37, 46, 55, 64, 73, 82, 91} 9

The set 3s10 has 54 three digit stifled numbers in base ten:

3s10 = {109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 19208, 217, 226, 235, 244, 253, 262, 271, 280, 28307, 316, 325, 334, 343, 352, 361, 370, 36406, 415, 424, 433, 442, 451, 460, 43505, 514, 523, 532, 541, 550, 49604, 613, 622, 631, 640, 54703, 712, 721, 730, 58802, 811, 820, 61901, 910} 63

The set 4s10 has 218 four digit stifled numbers in base ten:

4s10 = {1009, 1018, 1027, 1036, 1045, 1054, 1063, 1072, 1081, 1090, 731108, 1117, 1126, 1135, 1144, 1153, 1162, 1171, 1280, 821207, 1216, 1225, 1234, 1243, 1252, 1261, 1270, 901306, 1315, 1324, 1333, 1342, 1351, 1360, 971405, 1414, 1423, 1432, 1441, 1450, 1031504, 1513, 1522, 1531, 1540, 1081603, 1612, 1621, 1630, 1121702, 1711, 1720, 1151801, 1810, 1171900, 1182008, 2017, 2026, 2035, 2044, 2053, 2062, 2071, 2080 127

2107, 2116, 2125, 2134, 2143, 2152, 2161, 2170, 1352206, 2213, 2224, 2233, 2242, 2251, 2260, 1422305, 2314, 2323, 2332, 2341, 2350, 1482404, 2413, 2422, 2431, 2440, 1532503, 2512, 2521, 2530, 1572602, 2611, 2620, 1602701, 2710, 1622800, 1633007, 3016, 3025, 3034, 3043, 3052, 3061, 3070, 1713106, 3015, 3024, 3033, 3042, 3051, 3060, 1783205, 3214, 3223, 3232, 3241, 3250, 1843304, 3313, 3322, 3331, 3340, 1893403, 3412, 3421, 3430, 193

3502, 3511, 3520, 1963601, 3610, 1983700, 1994006, 4015, 4024, 4033, 4042, 4051, 4060, 2064105, 4114, 4123, 4132, 4141, 4150, 2124204, 4213, 4222, 4231, 4240, 2174303, 4312, 4321, 4330, 2214402, 4411, 4420, 2244501, 4510, 2264600, 227


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60, 61, 62, 63,70, 71, 72,80, 81,90}

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Binary

In base 2, any number that is 2^a + 2^b is a stifled number where a and b arepositive integers and are not equal.

2^0 2^1 2^2 2^3 2^4 2^5 2^6 2^71 2 4 8 16 32 64 128 s2 nth

1 1 0 0 0 0 0 0 3 11 0 1 0 0 0 0 0 5 20 1 1 0 0 0 0 0 6 31 0 0 1 0 0 0 0 9 40 1 0 1 0 0 0 0 10 50 0 1 1 0 0 0 0 12 61 0 0 0 1 0 0 0 17 70 1 0 0 1 0 0 0 18 80 0 1 0 1 0 0 0 20 9

0 0 0 1 1 0 0 0 24 101 0 0 0 0 1 0 0 33 110 1 0 0 0 1 0 0 34 120 0 1 0 0 1 0 0 36 130 0 0 1 0 1 0 0 40 140 0 0 0 1 1 0 0 48 151 0 0 0 0 0 1 0 65 160 1 0 0 0 0 1 0 66 170 0 1 0 0 0 1 0 68 180 0 0 1 0 0 1 0 72 190 0 0 0 1 0 1 0 80 200 0 0 0 0 1 1 0 96 211 0 0 0 0 0 0 1 129 22

0 1 0 0 0 0 0 1 130 230 0 1 0 0 0 0 1 132 240 0 0 1 0 0 0 1 136 250 0 0 0 1 0 0 1 144 260 0 0 0 0 1 0 1 160 270 0 0 0 0 0 1 1 192 28

Stifled Numbers (Part 2)

by Adi Cox. 18-3-13______________________________________________________________________________

Number Stifled set Stifled Numbers in their nx of bases respective bases Stifled

Numbers______________________________________________________________________________

1 {NULL} - 02 {NULL} - 0


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3 {2} 11 14 {NULL} - 05 {2,3} 101,12 26 {2} 110 17 {3,4} 21,13 28 {NULL} - 09 {2,5} 1001,14 210 {2} 1010 111 {3,6} 102,15 212 {2} 1100 113 {3,4,5,7} 111,31,23,16 414 {NULL} - 015 {3,8} 120,17 216 {6} 24 117 {2,5,9} 10001,32,18 318 {2} 10020 119 {3,4,7,10} 201,103,25,19 420 {2} 10100 121 {3,5,6,11} 210,41,33,1(10) 422 {4,8} 112,26 223 {12} 1(11) 124 {2} 11000 125 {4,7,9,13} 121,34,27,1(12) 426 {6} 42 1

27 {14} 1(13) 128 {4,10} 130,28 229 {3,5,8,15} 1002,104,35,1(14) 430 {NULL} - 031 {3,6,7,11,16} 1011,51,43,29,1(15) 532 {NULL} - 033 {2,3,5,9,17} 10001,1020,113,36,1(16) 534 {2,4,12} 10010,202,2(10) 335 {18} 1(17) 136 {2,8} 100100,44 237 {3,4,5,7,10,13,19} 1101,211,122,52,37,2(11),1(18) 738 {NULL} - 039 {3,20} 1110,1(19) 2

40 {2,4,14} 101000,220,2(12) 341 {4,6,9,11,21} 131,105,45,38,1(20) 542 {NULL} - 043 {7,8,15,22} 61,53,2(13),1(21) 444 {NULL} - 045 {3,5,12,23} 1200,140,39,1(22) 446 {6,10,26} 114,46,2(14) 347 {24} 1(23) 148 {2,25} 110000,1(24) 249 {4,9,13,17} 301,54,3(10),2(15) 450 {8} 62 151 {6,11,26} 123,47,1(25) 352 {4,18} 310,2(16) 2

53 {5,14,27} 203,3(11),1(26) 354 {NULL} - 055 {3,7,10,19,28} 2001,106,55,2(17),1(27) 556 {6,12} 132,48 257 {3,5,8,9,15,29} 2010,212,71,63,3(12),1(28) 658 {20} 2(18) 159 {30} 1(29) 160 {NULL} - 061 {5,6,7,11,13,16,21,31} 221,141,115,56,49,3(13),2(19),1(30) 862 {NULL} - 0


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63 {3,32} 2100,1(31) 264 {10,22} 64,2(20) 265 {2,5,9,17,33} 1000001,230,72,3(14),1(32) 566 {2,6,14} 1000010,150,4(10) 367 {4,7,12,23,34} 1003,124,57,2(21),1(33) 568 {2} 1000100 169 {18,35} 3(15),1(34) 270 {4,24} 1012,2(22) 271 {8,11,15,36} 107,65,4(11),1(35) 472 {2} 1001000 173 {4,7,9,10,13,19,25,37} 1021,133,81,73,58,3(16),2(23),1(36) 874 {NULL} - 075 {38} 1(37) 176 {4,6,16,26} 1030,204,4(12),2(24) 477 {5,20,39} 302,3(17),1(38) 378 {8,12} 116,66 279 {7,14,27,40} 142,59,2(25),1(39) 480 {2} 1010000 181 {5,6,11,17,21,41} 311,213,74,4(13),3(18)1(40) 682 {4,10,28} 1102,82,2(26) 383 {3,42} 10002,1(41) 284 {NULL} - 085 {3,4,5,7,8,13, 10011,1111,320,151,125,67,

15,22,29,43} 5(10),3(19),2(27),1(42) 10

86 {6,18} 222,4(14) 287 {3,44} 10020,1(43) 288 {4,30} 1120,2(28) 289 {9,12,23,45} 108,75,3(20),1(44) 490 {NULL} - 091 {3,6,7,10,11,16, 10101;231,160,91,83,5(11),

19,24,31,46} 4(15),3(21),2(29),1(45) 1092 {8,14} 134,68 293 {3,24,47} 10110,3(21),1(46) 394 {32} 2(30) 195 {48} 1(47) 196 {2,6,20} 1100000,240,4(16) 397 {4,9,13,17,25,33,49} 1201,117,76,5(12),3(22),2(31),1(48) 7

98 {NULL} - 099 {3,8,15,50} 10200,143,69,1(49) 4100 {4,12,34} 1210,84,2(32) 3

__________________________________________________________________________

The html file below was used to find the stifled numbers in their variousbases above.(Change the value of B to change the base, which is currently set to binary)

__________________________________________________________________________

A for loop

document.write("The 6 digit base B numbers = B")


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(S-1) Stifled Minus One Numbers:These numbers add up to one less than their base.

by Adi Cox. 7-5-13______________________________________________________________________________

Number S-1 set S-1 Numbers in their nx of bases respective bases S-1

Numbers______________________________________________________________________________

1 {1} 0 12 {2,3} 10,2 23 {4} 3 14 {2,3,5} 100,11,4 35 {6} 5 16 {3,4,7} 20,12,6 37 {8} 7 18 {2,5,9} 1000,13,8 39 {4,10} 21,9 210 {3,6,11} 101,14,(10) 311 {12} (11) 112 {3,4,5,7,13} 110,30,22,15,(12) 5

13 {14} (13) 114 {8,15} 16,(14) 215 {6,16} 23,(15) 216 {2,5,9,17} 10000,31,17,(16) 417 {18} (17) 118 {3,4,7,10,19} 200,102,24,18,(18) 519 {20} (19) 120 {5,6,11,21} 40,32,19,(20) 421 {4,8,22} 111,25,(21) 322 {12,23} 1(10),(22) 223 {24} (23) 124 {4,7,9,13,25} 120,33,26,1(11),(24) 525 {6,26} 41,(25) 2

26 {14,27} 1(12),(26) 227 {10,28} 27,(27) 228 {3,5,8,15,29} 1001,103,34,1(13),(28) 529 {30} (29) 130 {3,6,7,11,16,31} 1010,50,42,28,1(14),(30) 631 {32} (31) 132 {2,5,9,17,33} 100000,112,35,1(15),(32) 533 {4,12,34} 201,29,(33) 334 {18,35} 1(16),(34) 235 {8,36} 43,(35) 236 {3,4,5,7,10,13,19,37 1100,210,121,51,36,2(10),1(17),(36) 837 {38} (37) 138 {20,39} 1(18),(38) 2

39 {14,40} 2(11),(39) 240 {5,6,9,11,21,41} 130,104,44,37,1(19),(41) 641 {42} (42) 142 {7,8,15,22,43} 60,52,2(12),1(20),(42) 544 {12,23,45} 38,1(21),(44) 345 {6,10,16,46} 113,45,2(13),(45) 446 {24,47} 1(22),(46) 247 {48} (47) 148 {4,9,13,17,25,49} 300,53,39,2(14),1(23),(48) 649 {8,50} 61,(49) 2


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50 {6,11,26,51} 122,46,1(24),(50) 451 {18,52} 2(15),(51) 252 {5.14,27,53} 202,3(10),1(25),(52) 453 {54} (53) 154 {3,7,10,19,28,55} 2000,105,54,2(16),1(26),(54) 655 {6,12,56} 131,47,(55) 356 {5,8,9,15,29,57} 211,70,62,3(11),1(27),(56) 657 {20,58} 2(17),(57) 258 {30,59} 1(28),(58) 259 {60} (59) 160 {5,6,7,11,13, 220,140,114,55,48,

16,21,31,61} 312,218,129,(60) 961 {62} (61) 162 {32,63} 1(30),(62) 263 {10,22,64} 63,2(19),(63) 364 {2,9,17,33,65} 1000000,71,3(13),1(31),(64) 565 {14,66} 49,(65) 266 {4,7,12,23,34,67} 1002,123,56,2(20),1(32),(66) 667 {68} (67) 168 {18,35,69} 3(14),1(33),(68) 369 {4,24,70} 1011,2(21),(69) 370 {11,15,36,71} 64,4(10),1(34),(70) 471 {72} 71 172 {4,7,9,10,13, 1020,132,80,72,57,

19,25,37,73} 3(15),2(22),1(35),(72) 973 {74} (73) 174 {38,75} 1(36),(74) 275 {6,16,26,76} 203,4(11),2(23),(75) 476 {5,20,39,77} 301,3(16),1(37),(76) 477 {8,12,78} 115,65,(77) 378 {7,14,27,40,79} 141,58,2(24),1(38),(78) 579 {80} (79) 180 {5,6,11,17,21,41,81} 310,212,73,4(12),3(17),1(39),(81) 781 {4,10,28,82} 1101,81,2(25),(82) 482 {3,42,83} 10001,1(40),(82) 383 {84} (83) 184 {3,4,7,8,13,15, 10010,1110,150,124,66,59,

22,29,43,85} 3(18),2(26),1(41),(84) 1085 {6,18,86} 221,4(13),(85) 386 {44,87} 1(42),(86) 287 {30} 2(27),(87) 288 {12,23,45,89} 74,3(19),1(43),(88) 489 {90} (89) 190 {3,6,10,11,16, 10100,230,90,82,5(10),

19,31,46,91} 4(14),2(28),1(44),(90) 991 {8,14,92} 133,67,(91) 392 {24,47,93} 3(20),1(45),(92) 393 {32,94} 2(29),(93) 294 {48,95} 1(46),(94) 295 {20,96} 4(15),(95) 2

96 {4,13,17,25,33,49.97} 1200,75,5(11),3(21),2(30),1(47),(96) 797 {98} (97) 198 {8,15,50} 142,68,1(48) 399 {12,34} 83,2(31) 2100 {5,11,51} 400,91,1(49) 3


23/126

_______________________________________________________________________________

Finding Fractions In Other Bases

by Adi Cox 17-7-14

_______________________________________________________________________________

1/19 In Base 2

First find 19 in base 2:

19/2 = 9 rem 19/2 = 4 rem 14/2 = 2 rem 02/2 = 1 rem 01/2 = 0 rem 1

so 19 = 10011

Check the answer:

16 8 4 2 1___________________________________

1 0 0 1 1

16 + 0 + 0 + 2 + 1 = 19

Long division in binary:

. . 0.00001101011110010100001 __________________________10011 | 1.00000000000000000000000

10011 11010 1110 11100 10011 10010

100100 10011 100010

100111111010011 10110 10011 110 1100


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11000 10011

100100010000

100000 10011

. .0.0000110101111001010000110101111001010000110101111001010000110101111001010000

1/1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 ..._______________________________________________________________________________

0. 0 0 0 0 1 1 0 1 ...

If we add up the first 9 one digits in the infinite binary power series for 1/19we get:

1/32+1/64+1/256+1/1024+1/2048+1/4096+1/8192+1/65536+1/262144 = 0.051166574

where 1/19 = 0.052631579

Using an iterative formula to find 1/19 In Base 2 I have written a program usinghtml.

To change the base in the html script below change the value of aand to change the fraction change the value of b.

_______________________________________________________________________________

A for loop

document.write("In base a, one divided by b gives: 0.x1,x2,x3,... xn.")

a=2;document.write("a = ", a ,"
");

b=19;document.write("b = ", b ,"

");


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c1=a%b;

d1=a/b;

x1=d1-(c1/b)document.write("x1 = ", x1 ,"
");

c2=(c1*a)%b

d2=(c1*a)/b

");

c3=(c2*a)%b

d3=(c2*a)/b

");

c4=(c3*a)%b

d4=(c3*a)/b

");

c5=(c4*a)%b

d5=(c4*a)/b

");

c6=(c5*a)%b

d6=(c5*a)/b

");

c7=(c6*a)%b

d7=(c6*a)/b

");


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c8=(c7*a)%b

d8=(c7*a)/b

");

c9=(c8*a)%b

d9=(c8*a)/b

");

c10=(c9*a)%b

d10=(c9*a)/b

x10=d10-(c10/b)

document.write("x10 = ", x10 ,"
");

c11=(c10*a)%b;

d11=(c10*a)/b;

");

c12=(c11*a)%b

d12=(c11*a)/b

");

c13=(c12*a)%b

d13=(c12*a)/b

");

c14=(c13*a)%b

d14=(c13*a)/b


27/126

");

c15=(c14*a)%b

d15=(c14*a)/b

");

c16=(c15*a)%b

d16=(c15*a)/b

");

c17=(c16*a)%b

d17=(c16*a)/b

");

c18=(c17*a)%b

d18=(c17*a)/b

");

c19=(c18*a)%b

d19=(c18*a)/b

");

c20=(c19*a)%b

d20=(c19*a)/b

");

c21=(c20*a)%b

d21=(c20*a)/b


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");

c22=(c21*a)%b

d22=(c21*a)/b

");

c23=(c22*a)%b

d23=(c22*a)/b

");

c24=(c23*a)%b

d24=(c23*a)/b

");

c25=(c24*a)%b

d25=(c24*a)/b

");

c26=(c25*a)%b

d26=(c25*a)/b

");

c27=(c26*a)%b

d27=(c26*a)/b

");

c28=(c27*a)%b


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d28=(c27*a)/b

");

c29=(c28*a)%b

d29=(c28*a)/b

");

c30=(c29*a)%b

d30=(c29*a)/b

");

c31=(c30*a)%b

d31=(c30*a)/b

");

c32=(c31*a)%b

d32=(c31*a)/b

");

c33=(c32*a)%b

d33=(c32*a)/b

");

c34=(c33*a)%b

d34=(c33*a)/b

");


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c35=(c34*a)%b

d35=(c34*a)/b

");

c36=(c35*a)%b

d36=(c35*a)/b

");

c37=(c36*a)%b

d37=(c36*a)/b

");

c38=(c37*a)%b

d38=(c37*a)/b

");

c39=(c38*a)%b

d39=(c38*a)/b

");

c40=(c39*a)%b

d40=(c39*a)/b

");

c41=(c40*a)%b

d41=(c40*a)/b

");


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32/126

c49=(c48*a)%b

d49=(c48*a)/b

");

c50=(c49*a)%b

d50=(c49*a)/b

");

_______________________________________________________________________________

The output of the above html program is below

_______________________________________________________________________________

In base a, one divided by b gives: 0.x1,x2,x3,... xn.a = 2b = 19

x1 = 0x2 = 0x3 = 0x4 = 0x5 = 0.9999999999999999x6 = 1x7 = 0x8 = 1x9 = 0x10 = 0.9999999999999999x11 = 1x12 = 1x13 = 1

x14 = 0x15 = 0x16 = 1x17 = 0x18 = 1x19 = 0x20 = 0x21 = 0x22 = 0x23 = 0.9999999999999999


33/126


34/126

1/1 0.11111111111111111111111111111111111111111111111111111111111111111111111/2 0.10000000000000000000000000000000000000000000000000000000000000000000001/3 0.01010101010101010101010101010101010101010101010101010101010101010101011/4 0.01000000000000000000000000000000000000000000000000000000000000000000001/5 0.00110011001100110011001100110011001100110011001100110011001100110011001/6 0.00101010101010101010101010101010101010101010101010101010101010101010101/7 0.00100100100100100100100100100100100100100100100100100100100100100100101/8 0.00100000000000000000000000000000000000000000000000000000000000000000001/9 0.00011100011100011100011100011100011100011100011100011100011100011100011/10 0.00011001100110011001100110011001100110011001100110011001100110011001101/11 0.00010111010001011101000101110100010111010001011101000101110100010111011/12 0.00010101010101010101010101010101010101010101010101010101010101010101011/13 0.00010011101100010011101100010011101100010011101100010011101100010011101/14 0.00010010010010010010010010010010010010010010010010010010010010010010011/15 0.00010001000100010001000100010001000100010001000100010001000100010001001/16 0.00010000000000000000000000000000000000000000000000000000000000000000001/17 0.00001111000011110000111100001111000011110000111100001111000011110000111/18 0.00001110001110001110001110001110001110001110001110001110001110001110001/19 0.00001101011110010100001101011110010100001101011110010100001101011110011/20 0.00001100110011001100110011001100110011001100110011001100110011001100111/21 0.00001100001100001100001100001100001100001100001100001100001100001100001/22 0.00001011101000101110100010111010001011101000101110100010111010001011101/23 0.00001011001000010110010000101100100001011001000010110010000101100100001/24 0.0000101010101010101010101010101010101010101010101010101010101010101010

1/25 0.00001010001111010111000010100011110101110000101000111101011100001010001/26 0.00001001110110001001110110001001110110001001110110001001110110001001111/27 0.00001001011110110100001001011110110100001001011110110100001001011110111/28 0.00001001001001001001001001001001001001001001001001001001001001001001001/29 0.00001000110100111101110010110000100011010011110111001011000010001101001/30 0.00001000100010001000100010001000100010001000100010001000100010001000101/31 0.00001000010000100001000010000100001000010000100001000010000100001000011/32 0.00001000000000000000000000000000000000000000000000000000000000000000001/33 0.00000111110000011111000001111100000111110000011111000001111100000111111/34 0.00000111100001111000011110000111100001111000011110000111100001111000011/35 0.00000111010100000111010100000111010100000111010100000111010100000111011/36 0.00000111000111000111000111000111000111000111000111000111000111000111001/37 0.0000011011101011001111100100010100110000011011101011001111100100010100

1/38 0.00000110101111001010000110101111001010000110101111001010000110101111001/39 0.00000110100100000110100100000110100100000110100100000110100100000110101/40 0.00000110011001100110011001100110011001100110011001100110011001100110011/41 0.00000110001111100111000001100011111001110000011000111110011100000110001/42 0.00000110000110000110000110000110000110000110000110000110000110000110001/43 0.00000101111101000001011111010000010111110100000101111101000001011111011/44 0.00000101110100010111000101110100010111010001011101000101110100010111011/45 0.00000101101100000101101100000101101100000101101100000101101100000101101/46 0.00000101100100001011001000010110010000101100100001011001000010110010001/47 0.00000101011100100110001000001010111001001100010000010101110010011000101/48 0.00000101010101010101010101010101010101010101010101010101010101010101011/49 0.00000101001110010111100000101001110010111100000101001110010111100000101/50 0.0000010100011110101110000101000111101011100001010001111010111000010100

FRACTIONS 1/x IN BASE 3

1/1 0.22222222222222222222222222222222222222222222222222222222222222222222221/2 0.11111111111111111111111111111111111111111111111111111111111111111111111/3 0.10000000000000000000000000000000000000000000000000000000000000000000001/4 0.02020202020202020202020202020202020202020202020202020202020202020202021/5 0.01210121012101210121012101210121012101210121012101210121012101210121011/6 0.0111111111111111111111111111111111111111111111111111111111111111111111


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1/7 0.01021201021201021201021201021201021201021201021201021201021201021201021/8 0.01010101010101010101010101010101010101010101010101010101010101010101011/9 0.01000000000000000000000000000000000000000000000000000000000000000000001/10 0.00220022002200220022002200220022002200220022002200220022002200220022001/11 0.00211002110021100211002110021100211002110021100211002110021100211002111/12 0.00202020202020202020202020202020202020202020202020202020202020202020201/13 0.00200200200200200200200200200200200200200200200200200200200200200200201/14 0.00122100122100122100122100122100122100122100122100122100122100122100121/15 0.00121012101210121012101210121012101210121012101210121012101210121012101/16 0.00120012001200120012001200120012001200120012001200120012001200120012001/17 0.00112021221102010011202122110201001120212211020100112021221102010011201/18 0.00111111111111111111111111111111111111111111111111111111111111111111111/19 0.00110210022112012200110210022112012200110210022112012200110210022112011/20 0.00110011001100110011001100110011001100110011001100110011001100110011001/21 0.00102120102120102120102120102120102120102120102120102120102120102120101/22 0.00102001020010200102001020010200102001020010200102001020010200102001021/23 0.00101120021001011200210010112002100101120021001011200210010112002100101/24 0.00101010101010101010101010101010101010101010101010101010101010101010101/25 0.00100201102212202112001002011022122021120010020110221220211200100201101/26 0.00100100100100100100100100100100100100100100100100100100100100100100101/27 0.00100000000000000000000000000000000000000000000000000000000000000000001/28 0.00022200022200022200022200022200022200022200022200022200022200022200021/29 0.00022101020111222001212021110002210102011122200121202111000221010201111/30 0.0002200220022002200220022002200220022002200220022002200220022002200220

1/31 0.00021211122102022201011100120200021211122102022201011100120200021211121/32 0.00021121000211210002112100021121000211210002112100021121000211210002111/33 0.00021100211002110021100211002110021100211002110021100211002110021100211/34 0.00021010222012120002101022201212000210102220121200021010222012120002101/35 0.00020221110100020221110100020221110100020221110100020221110100020221111/36 0.00020202020202020202020202020202020202020202020202020202020202020202021/37 0.00020120022202102200020120022202102200020120022202102200020120022202101/38 0.00020101122202121100020101122202121100020101122202121100020101122202121/39 0.00020020020020020020020020020020020020020020020020020020020020020020021/40 0.00020002000200020002000200020002000200020002000200020002000200020002001/41 0.00012221000122210001222100012221000122210001222100012221000122210001221/42 0.00012210012210012210012210012210012210012210012210012210012210012210011/43 0.0001212212020021112102221010010202201110120001212212020021112102221010

1/44 0.00012112010001211201000121120100012112010001211201000121120100012112011/45 0.00012101210121012101210121012101210121012101210121012101210121012101211/46 0.00012021122000120211220001202112200012021122000120211220001202112200011/47 0.00012011121002022001011000120111210020220010110001201112100202200101101/48 0.00012001200120012001200120012001200120012001200120012001200120012001201/49 0.00011221220020121112022211001002202101110200011221220020121112022211001/50 0.0001121201222110102100011212012221101021000112120122211010210001121201


1/1 0.33333333333333333333333333333333333333333333333333333333333333333333331/2 0.2000000000000000000000000000000000000000000000000000000000000000000000

1/3 0.11111111111111111111111111111111111111111111111111111111111111111111111/4 0.10000000000000000000000000000000000000000000000000000000000000000000001/5 0.03030303030303030303030303030303030303030303030303030303030303030303031/6 0.02222222222222222222222222222222222222222222222222222222222222222222221/7 0.02102102102102102102102102102102102102102102102102102102102102102102101/8 0.02000000000000000000000000000000000000000000000000000000000000000000001/9 0.01301301301301301303013013013013013013013013013013013030130130130130131/10 0.01212121212121212121212121212121212121212121212121212121212121212121211/11 0.01131011310113101131011310113101131011310113101131011310113101131011311/12 0.0111111111111111111111111111111111111111111111111111111111111111111111


36/126

1/13 0.01032301032301032301032301032301032301032301032301032301032301032301031/14 0.01021021021021021021021021021021021021021021021021021021021021021021021/15 0.01010101010101010101010101010101010101010101010101010101010101010101011/16 0.01000000000000000000000000000000000000000000000000000000000000000000001/17 0.00330033003300330033003300330033003300330033003300330033003300330033001/18 0.00320320320320320320320320320320320320320320320320320320320320320320321/19 0.00311321100311321100311321100311321100311321100311321100311321100311321/20 0.00303030303030303030303030303030303030303030303030303030303030303030301/21 0.00300300300300300300300300300300300300300300300300300300300300300300301/22 0.00232202322023220232202322023220232202322023220232202322023220232202321/23 0.00230201121002302011210023020112100230201121002302011210023020112100231/24 0.00222222222222222222222222222222222222222222222222222222222222222222221/25 0.00220331130022033113002203311300220331130022033113002203311300220331131/26 0.00213120213120213120213120213120213120213120213120213120213120213120211/27 0.00211323100211323100211323100211323100211323100211323100211323100211321/28 0.00210210210210210210210210210210210210210210210210210210210210210210211/29 0.00203103313023002031033130230020310331302300203103313023002031033130231/30 0.00202020202020202020202020202020202020202020202020202020202020202020201/31 0.00201002010020100201002010020100201002010020100201002010020100201002011/32 0.00200000000000000000000000000000000000000000000000000000000000000000001/33 0.00133001330013300133001330013300133001330013300133001330013300133001331/34 0.00132013201320132013201320132013201320132013201320132013201320132013201/35 0.00131100131100131100131100131100131100131100131100131100131100131100131/36 0.0013013013013013013013013013013013013013013013013013013013013013013013

1/37 0.00123223033210110300123223033210110300123223033210110300123223033210111/38 0.00122330220122330220122330220122330220122330220122330220122330220122331/39 0.00122100122100122100122100122100122100122100122100122100122100122100121/40 0.00121212121212121212121212121212121212121212121212121212121212121212121/41 0.00120332130012033213001203321300120332130012033213001203321300120332131/42 0.00120120120120120120120120120120120120120120120120120120120120120120121/43 0.00113310011331001133100113310011331001133100113310011331001133100113311/44 0.00113101131011310113101131011310113101131011310113101131011310113101131/45 0.00112300112300112300112300112300112300112300112300112300112300112300111/46 0.00112100230201121002302011210023020112100230201121002302011210023020111/47 0.00111302120200223210301001113021202002232103010011130212020022321030101/48 0.00111111111111111111111111111111111111111111111111111111111111111111111/49 0.0011032113200221302330011032113200221302330011032113200221302330011032

1/50 0.0011013223201101322320110132232011013223201101322320110132232011013223


1/1 0.44444444444444444444444444444444444444444444444444444444444444444444441/2 0.22222222222222222222222222222222222222222222222222222222222222222222221/3 0.13131313131313131313131313131313131313131313131313131313131313131313131/4 0.11111111111111111111111111111111111111111111111111111111111111111111111/5 0.10000000000000000000000000000000000000000000000000000000000000000000001/6 0.04040404040404040404040404040404040404040404040404040404040404040404041/7 0.0324120324120324120324120324120324120324120324120324120324120324120324

1/8 0.03030303030303030303030303030303030303030303030303030303030303030303031/9 0.02342102342102342102342102342102342102342102342102342102342102342102341/10 0.02222222222222222222222222222222222222222222222222222222222222222222221/11 0.02114021140211402114021140211402114021140211402114021140211402114021141/12 0.02020202020202020202020202020202020202020202020202020202020202020202021/13 0.01430143014301430143014301430143014301430143014301430143014301430143011/14 0.01343101343101343101343101343101343101343101343101343101343101343101341/15 0.01313131313131313131313131313131313131313131313131313131313131313131311/16 0.01240124012401240124012401240124012401240124012401240124012401240124011/17 0.0121340243231042012134024323104201213402432310420121340243231042012134


37/126

1/18 0.01143301143301143301143301143301143301143301143301143301143301143301141/19 0.01124214101124214101124214101124214101124214101124214101124214101124211/20 0.01111111111111111111111111111111111111111111111111111111111111111111111/21 0.01043401043401043401043401043401043401043401043401043401043401043401041/22 0.01032010320103201032010320103201032010320103201032010320103201032010321/23 0.01020413321434240311230102041332143424031123010204133214342403112301021/24 0.01010101010101010101010101010101010101010101010101010101010101010101011/25 0.01000000000000000000000000000000000000000000000000000000000000000000001/26 0.00440044004400440044004400440044004400440044004400440044004400440044001/27 0.00430332244014112200430332244014112200430332244014112200430332244014111/28 0.00421300421300421300421300421300421300421300421300421300421300421300421/29 0.00412334403211004123344032110041233440321100412334403211004123344032111/30 0.00404040404040404040404040404040404040404040404040404040404040404040401/31 0.00400400400400400400400400400400400400400400400400400400400400400400401/32 0.00342312003423120034231200342312003423120034231200342312003423120034231/33 0.00334322030033432203003343220300334322030033432203003343220300334322031/34 0.00331423441130210033142344113021003314234411302100331423441130210033141/35 0.00324120324120324120324120324120324120324120324120324120324120324120321/36 0.00321400321400321400321400321400321400321400321400321400321400321400321/37 0.00314212204011334244130232240433110200314212204011334244130232240433111/38 0.00312104300312104300312104300312104300312104300312104300312104300312101/39 0.00310031003100310031003100310031003100310031003100310031003100310031001/40 0.00303030303030303030303030303030303030303030303030303030303030303030301/41 0.0030110220441433422400301102204414334224003011022044143342240030110220

1/42 0.00244200244200244200244200244200244200244200244200244200244200244200241/43 0.00242314122343404311144202130322101040133300242314122343404311144202131/44 0.00241002410024100241002410024100241002410024100241002410024100241002411/45 0.00234210234210234210234210234210234210234210234210234210234210234210231/46 0.00232431410442120130340023243141044212013034002324314104421201303400231/47 0.00231221041114241401012442132234033302030434320023122104111424140101241/48 0.00230023002300230023002300230023002300230023002300230023002300230023001/49 0.00223341432133140424344221103012311304020100223341432133140424344221101/50 0.0022222222222222222222222222222222222222222222222222222222222222222222


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_____________________________________________________________________

Investigating The Iterative Formula:

x = kx(1 - x)

by Adi Cox 11/8/14_____________________________________________________________________

where k = 4 and the starting value of x = 0.2we get chaotic fractions:

x0 = (4/5)(4/5) = 16/25 = (2^4)/(5^2) = 0.64

x1 = (64/25)(9/25) = 576/625 = (2^6*3^2)/(5^4) = ((2^6)*3^2)/(5^4) = 0.9216

x2 = (2304/625)(49/625)

= 112896/390625 = (2^8*3^2*7^2)/(5^8) = ((2^8)*(3*7)^2)/(5^8) = 0.28901376

x3 = (451584/390625)(277729/390625) = 125417972736/152587890625 = (2^10*3^2*7^2*17^2*31^2)/(5^16) = ((2^10)*(3*7*17*31)^2)/(5^16) = 0.8219392261226496

x4 = (501671890944/152587890625)(27169917889/152587890625) = 13630384084167842697216/23283064365386962890625

= (2^12*3^2*7^2*17^2*31^2*191^2*863^2)/(5^32) = ((2^12)*(3*7*17*31*191*863)^2)/(5^32) = 0.58542053873419795759415150247936

xn = [2^(2+2n)*MnValuesSquared]/[5^(2^n)]

How do we find the M values?

nth M value {1, 2, 3, 4 }nth prime {2, 4, 7,11, 43,150 }primes {3, 7, 17*31, 191*863 }

M values {3, 7, 527, 164833 }

Below are the list of M valuesM0 ---> 0M1 ---> 3M2 ---> 7M3 ---> 527M4 ---> 164833


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M5 ---> ?_____________________________________________________________________

The chaotic fraction for x5

xn = [2^(2+2n)*MnValuesSquared]/[5^(2^n)]

x5 = [2^(2+2*6)*(*M1*M2*M3*M4*M5)^2]/[5^(2^6)] = [2^(2+2*6)*(3*7*527*164833*M5)^2]/[5^(2^6)] = [73786976294838206464*(1824206811*M5)^2]/[5^64] = 0.970813326249439

M5 = {([0.970813326249439*5^(2^n)]/[2^(2+2*6)])^(1/2)}/{M1*M2*M3*M4}_____________________________________________________________________

The html file below gives the output for the first 1000 values forthe iterative formula x = kx(1 - x)

where k = 4 and the starting value of x = 0.2

For some reason the answers are slightly wrong. The first an

00 Mathematics by Adi Cox 00

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