Mathematics by Adi Cox

8/9/2019 Mathematics by Adi Cox

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_____________________________________________________________________

What Is An AdiPolytope?

written by Adi Cox 1st April 2015

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There is a video on youtube that ao!panies this paper"

https"##www$youtube$o!#wath?v%&CA'1(PbC)*

Introdution

____________

https://www.youtube.com/watch?v=qCAB1HPbC-Yhttps://www.youtube.com/watch?v=qCAB1HPbC-Y


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To +nd out what an Adi polytope is we need to understand what a

!easure and a si!plex is$ This is beause an Adi polytope is a

!easure with so!e ri,ht si!plexes ta-en away .ro! it$

There is an Adi polytope in every di!ension that is 2 or above$ This

is beause there are no ri,ht si!plexes in less than 2 di!ensional

spae$ It does ,et on.usin, beause the 2nd di!ensional Adi

Polytope is a one di!ensional line$

An explination o. the notation used"

xy/ xy/ xy/ xy/

000 )) 0013 0103 1004

point 000 is onneted to points 0013 010 and 100$

where xy/ are the axis in spae$

What Is A easure?

__________________

There is a !easure in every di!ension$ We an use binary to +nd

!easures in any di!ension"


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0th di!ension !easure is the point"

0

1st di!ension !easure is the line"

0 )) 1

2nd di!ension !easure is the s&uare"

00 )) 103 014

01 )) 113 004

10 )) 003 114

11 )) 013 104

6rd di!ension !easure is the ube"


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000 )) 1003 0103 0014

001 )) 1013 0113 0004

010 )) 1103 0003 0114

011 )) 1113 0013 0104

100 )) 0003 1103 1014

101 )) 0013 1113 1004

110 )) 0103 1003 1114

111 )) 0113 1013 1104

7th di!ension !easure is the tesserat"


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0000 )) 10003 01003 00103 00014

0001 )) 10013 01013 00113 00004

0010 )) 10103 01103 00003 00114

0011 )) 10113 01113 00013 00104

0100 )) 10003 00003 01103 01014

0101 )) 11013 00013 01113 01004

0110 )) 11103 00103 01003 01114

0111 )) 11113 00113 01013 01104


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1000 )) 00003 11003 10103 10014

1001 )) 00013 11013 10113 10004

1010 )) 00103 11103 10003 10114

1011 )) 00113 11113 10013 10104

1100 )) 01003 10003 11103 11014

1101 )) 01013 10013 11113 11004

1110 )) 01103 10103 11003 11114

1111 )) 01113 10113 11013 11104

What Is A 8i!plex?

__________________

A si!plex has all points onneted$ In n di!ensions the si!plex in

that di!ension has n91 points$ :sin, binary we an loo- at the

re,ular si!plex and the ri,ht si!plex"

0th di!ension si!plex is the point"

0

1st di!ension si!plex is the line"

0 )) 1


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2nd di!ension si!plex is"

00 )) 013 104

01 )) 103 004

10 )) 003 014

Above is a ri,ht si!plex beause there is a ri,ht an,le at point 00$

6rd di!ension si!plex is"

000 )) 0013 0103 1004

001 )) 0003 0103 1004

010 )) 0003 0013 1004

100 )) 0003 0013 0104



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100 )) 0103 0013 1114

010 )) 1003 0013 1114

001 )) 0103 0013 1114

111 )) 1003 0103 0014

Above is a re,ular si!plex beause all side len,ths are e&ual

7th di!ension si!plex is"

0000 )) 00013 00103 01003 10004

0001 )) 00003 00103 01003 10004

0010 )) 00003 00013 01003 10004

0100 )) 00003 00013 00103 10004

1000 )) 00003 00103 01003 00014



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1000 )) 01003 00103 00013 11114

0100 )) 10003 00103 00013 11114

0010 )) 10003 01003 00013 11114

0001 )) 10003 00103 00013 11114

1111 )) 10003 01003 00103 00014

Above is a re,ular si!plex beause all side len,ths are e&ual

8o what is an Adi Polytope?

___________________________

An Adi Polytope is an n di!ensional !easure with 2;$,$ The 2nd di!ension is a trivial exa!ple"

8tart with a s&uare and ta-in, away two ri,ht an,le trian,les we ,et

ust the strai,ht line where the hypotinuse o. both ri,ht an,le


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trian,les !eet$ 8o in the seond di!ension the Adi Polytope is ust

a strai,ht line$

2nd di!ension Adi Polytope is"

2nd di!ension !easure !inus 2nd di!ension ri,ht si!plexes 1 and 2

2nd di!ension !easure"

00 )) 103 014

01 )) 113 004

10 )) 003 114

11 )) 013 104

2nd di!ension ri,ht si!plex 1"

00 )) 013 104

01 )) 103 004

10 )) 003 014

2nd di!ension ri,ht si!plex 2"

11 )) 013 104

01 )) 103 114

10 )) 113 014


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8o when we ta-e away both o. these ri,ht si!plexes we are le.t with

nothin,3 everythin, is anelled out in the seond di!ension3 but we

,et a 1 di!ensional line$ This line is .ro! the points 01 )) 10

and this is the +rst Adi polytope$

6rd di!ension Adi Polytope is"

6rd di!ension !easure !inus 6rd di!ension si!plexes 13236 and 7

6rd di!ension !easure"

000 )) 1003 0103 0014

001 )) 1013 0113 0004

010 )) 1103 0003 0114

011 )) 1113 0013 0104

100 )) 0003 1103 1014

101 )) 0013 1113 1004


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110 )) 0103 1003 1114

111 )) 0113 1013 1104

6rd di!ension ri,ht si!plex 1 is"

000 )) 0013 0103 1004

001 )) 0003 0103 1004

010 )) 0003 0013 1004

100 )) 0003 0013 0104


100 )) 0103 1103 1114

010 )) 1003 1103 1114

011 )) 1003 0103 1114

111 )) 1003 0103 1104


100 )) 0013 1013 1114

001 )) 1003 1013 1114

101 )) 1003 0013 1114

111 )) 1003 0013 1014



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010 )) 0013 0113 1114

001 )) 0103 0113 1114

011 )) 0103 0013 1114

111 )) 0103 0013 0114

8o when we ta-e away these .our 6 di!ensional ri,ht si!plexes .ro!

the 6 di!ensional !easure we ,et a re,ular si!plex$ In the lan,ua,e

o. the 6rd di!ension$ We ta-e away .our ri,ht tetrahedrons .ro! a

ube and we are le.t with a re,ular tetrahedron as the Adi Polytope

o. the 6rd di!ension$

@e,ular 6rd di!ension si!plex % @e,ular Tetrahedron % Adi 6 Polytope

100 )) 0103 0013 1114

010 )) 1003 0013 1114

001 )) 1003 0103 1114

111 )) 1003 0103 0014

I. we do the sa!e as above in 7th and 5th di!ensional spae we ,et the

.ollowin, Adi Polytopes respetively"


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The Quartic Polynomial, The Quintic Polynomial And Its Unsolvability

_____________________________________________________________________

Written By Adi Cox

11th May 2015

It is intriguing to learn that the quintic polynomial is not alwayssolvable by radicals. So I have looked into the structure ofpolynomials to see what I can find:

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Quartic Polynomial

_____________________________________________________________________

Ax^4 + Bx^3 + Cx^2 + Dx^1 + Ex^0 = 0

A = 1B = a+b+c+dC = ab+ac+ad+bc+bd+cd

D = abc+abd+acd+bcdE = abcd

where {-a,-b,-c,-d} are the four solutions to the quartic polynomial.

i | | | -1 ------------0------------ 1 | |

| -i

Example: where the solutions are: {-1, i, i, 1}

we change the signs:{a=1, b=-i, c=-i, d=-1}

A = 1

B = (1)+(-i)+(-i)+(-1) = -2i

C = (1)(-i)+(1)(-i)+(1)(-1)+(-i)(-i)+(-i)(-1)+(-i)(-1) = -i - i - 1 - 1 + i + i = -2

D = (1)(-i)(-i)+(1)(-i)(-1)+(1)(-i)(-1)+(-i)(-i)(-1) = -1 + i + i + 1 = 2i

E = (1)(-i)(-i)(-1) = 1


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And so

x^4 - 2ix^3 - 2x^2 + 2ix + 1 = 0

=> (x+1)(x-i)(x-i)(x-1)

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Quintic Polynomial

_____________________________________________________________________

Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex^1 + Fx^0 = 0

A = 1B = a+b+c+d+eC = ab+ac+ad+ae+bc+bd+be+cd+ce+deD = abc+abd+abe+acd+ace+ade+bcd+bce+bde+cdeE = abcd+abce+abde+acde+bcdeF = abcde

where {-a,-b,-c,-d,-e} are the five solutions to the quinticpolynomial.

y= 1 ^ | | -1 --0----2--3----5----7--------11--> x | |

y= -1

Example: where the solutions are: {2, 3, 5, 7, 11}

we change the signs:{a=-2, b=-3, c=-5, d=-7, e=-11}

A = 1

B = (-2)+(-3)+(-5)+(-7)+(-11) = -28

C = (-2)(-3)+(-2)(-5)+(-2)(-7)+(-2)(-11)+(-3)(-5)+(-3)(-7)+(-3)(-11) +(-5)(-7)+(-5)(-11)+(-7)(-11) = 6 + 10 + 14 + 22 + 15 + 21 + 33 + 35 + 55 + 77 = 288

D = (-2)(-3)(-5)+(-2)(-3)(-7)+(-2)(-3)(-11)+(-2)(-5)(-7) +(-2)(-5)(-11)+(-2)(-7)(-11)+(-3)(-5)(-7)+(-3)(-5)(-11) +(-3)(-7)(-11)+(-5)(-7)(-11) = - 30 - 42 - 66 - 70 - 110 - 154 - 105 - 165 - 231 - 385 = -1358


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E = (-2)(-3)(-5)(-7)+(-2)(-3)(-5)(-11)+(-2)(-3)(-7)(-11) +(-2)(-5)(-7)(-11)+(-3)(-5)(-7)(-11) = 210 + 330 + 462 + 770 + 1155 = 2927

F = (-2)(-3)(-5)(-7)(-11) = -2310

And so

x^5 - 28x^4 + 288x^3 - 1358x^2 + 2927x^1 - 2310x^0 = 0

=> (x-2)(x-3)(x-5)(x-7)(x-11)

_____________________________________________________________________

So What Happens With x^5 - x^1 + 1 = 0

The above quintic polynomial is unsolvable by radicals, so let ussee what we get when we apply the previous methods for solvingpolynomials:

Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex^1 + Fx^0 = 0

A = 1B = a+b+c+d+e = 0C = ab+ac+ad+ae+bc+bd+be+cd+ce+de = 0D = abc+abd+abe+acd+ace+ade+bcd+bce+bde+cde = 0E = abcd+abce+abde+acde+bcde = -1F = abcde = 1

where {-a,-b,-c,-d,-e} are the five solutions to the quinticpolynomial.

x^5 - x^1 + 1 = 0

=> 1x^5 - 1x^1 + 1x^0 = 0

=> (1)x^5 + (0)x^4 + (0)x^3 + (0)x^2 + (-1)x^1 + (1)x^0 = 0

So the problem above is going to take some satisfying. Nothing shortof inventing a new type of number with specific properties tosatisfy this puzzle.

I'M WORKING ON IT!


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Adi 5 Space Polytope

The following sixteen 5 space right simplexes are taken from the 5 space

measure to get an Adi 5 Space Polytope.

5RS00000 {00000!" 0000#" 000#0" 00#00" 0#000" #0000$

5RS000## {000##!" 0000#" 000#0" 00###" 0#0##" #00##$


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5RS00#0# {00#0#!" 0000#" 00#00" 00###" 0##0#" #0#0#$

5RS00##0 {00##0!" 000#0" 00#00" 00###" 0###0" #0##0$

5RS0#00# {0#00#!" 0000#" 0#000" 0#0##" 0##0#" ##00#$

5RS0#0#0 {0#0#0!" 000#0" 0#000" 0#0##" 0###0" ##0#0$

5RS0##00 {0##00!" 00#00" 0#000" 0##0#" 0###0" ###00$

5RS0#### {0####!" 00###" 0#0##" 0##0#" 0###0" #####$

5RS#000# {#000#!" 0000#" #0000" #00##" #0#0#" ##00#$

5RS#00#0 {#00#0!" 000#0" #0000" #00##" #0##0" ##0#0$

5RS#0#00 {#0#00!" 00#00" #0000" #0#0#" #0##0" ###00$

5RS#0### {#0###!" 00###" #00##" #0#0#" #0##0" #####$

5RS##000 {##000!" 0#000" #0000" ##00#" ##0#0" ###00$

5RS##0## {##0##!" 0#0##" #00##" ##00#" ##0#0" #####$

5RS###0# {###0#!" 0##0#" #0#0#" ##00#" ###00" #####$

5RS####0 {####0!" 0###0" #0##0" ##0#0" ###00" #####$

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document.write@h9!Adi 5 Polytope


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if@x9==0{a9=x9D#$else{a9=0$

if@xC==0{aC=xCD#$else{aC=0$

if@x#==0{a#=x#D#$else{a#=0$

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Adi 5 Polytope<

00000 { #0000 " 0#000 " 00#00 " 000#0 " 0000# $

000## { #00## " 0#0## " 00### " 0000# " 000#0 $

00#0# { #0#0# " 0##0# " 0000# " 00### " 00#00 $

00##0 { #0##0 " 0###0 " 000#0 " 00#00 " 00### $

0#00# { ##00# " 0000# " 0##0# " 0#0## " 0#000 $

0#0#0 { ##0#0 " 000#0 " 0###0 " 0#000 " 0#0## $

0##00 { ###00 " 00#00 " 0#000 " 0###0 " 0##0# $

0#### { ##### " 00### " 0#0## " 0##0# " 0###0 $

#000# { 0000# " ##00# " #0#0# " #00## " #0000 $

#00#0 { 000#0 " ##0#0 " #0##0 " #0000 " #00## $

#0#00 { 00#00 " ###00 " #0000 " #0##0 " #0#0# $

#0### { 00### " ##### " #00## " #0#0# " #0##0 $

##000 { 0#000 " #0000 " ###00 " ##0#0 " ##00# $

##0## { 0#0## " #00## " ##### " ##00# " ##0#0 $

###0# { 0##0# " #0#0# " ##00# " ##### " ###00 $

####0 { 0###0 " #0##0 " ##0#0 " ###00 " ##### $


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Number Dictionary18-4-15

by Adi Cox

_____________________________________________________________________

Forward

Numbers are my friends. if I was a number I would want to be happyand lucky. The prime number 367 is both lucky and happy but being aprime number means that it is deficient and it is stifled in base 7.

_____________________________________________________________________

A

Algebraic numberAny number that is the root of a non-zero polynomial with rationalcoefficients is an algebraic number.

AmicableThe smallest pair of amicable numbers is (220, 284); for the proper

divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, ofwhich the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71and 142, of which the sum is 220.{(220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), ...}

C

CompositeA natural number greater than 1 that is not a prime number is calleda composite number.

D

DecagonalThe n-th decagonal number is given by the formula d = 4n^2 - 3n. Sothe 5th decagonal number is: 4[5]^2-3[5] = 85{0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, ...}

DodecagonalThe dodecagonal number for n is given by the formula 5n^2 - 4n wheren > 0. So the 3rd dodecagonal is 5[3]^2-4[3] = 45-12 = 33{1, 12, 33, 64, 105, 156, 217, 288, 369, ...}

E

EvenAn even number is an integer of the form n = 2k, where k is aninteger.{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, ...}

F

FactorialThe nth factorial is the product of all n digits: 0!=1, 1!=1,2!=2x1=2, 3!=3x2x1=6, 4!=4x3x2x1=24, 5!=5x4x3x2x1=120, ...


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{1, 1, 2, 6, 24, 120, 720, ...}

FigurateA member of the subset of the sets above containing only triangularnumbers, square numbers, pentagonal numbers, hexagonal numbers,heptagonal numbers, octagonal numbers, nonagonal numbers, decagonalnumbers, dodecagonal numbers, pyramidal numbers, and their analogsin other dimensions.

FractionSee rational number.

FriendlyThe smallest friendly number is 6, forming for example the friendlypair 6 and 28 with abundancy s(6) / 6 = (1+2+3+6) / 6 = 2, the sameas s(28) / 28 = (1+2+4+7+14+28) / 28 = 2.

H

HappyA happy number goes to one when the digits of the numbers aresquared and added together:2^2+3^2=13 --> 1^2+3^2=10 --> 1^2+0^2=1{1, 7, 10, 13, 19, 23, 28, ...}

HarshadThe number 18 is a Harshad number in base 10, because the sum of thedigits 1 and 8 is 9 (1+8=9), and 18 is divisible by 9.7864 is a Harshad number in base 2: 7864=1111010111000. The sum ofthe digits equal 8 and 7864/8=983.

HeptagonalThe n-th heptagonal number is given by the formula (5n^2-3n)/2. Sothe 7th heptagonal number is: (5[7]^2-3[7])/2 = (5x49-21)/2 = 112{1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, ...}

Hexagonal

The formula for the nth hexagonal number is: 2n^2-n. So the 5thhexagonal number is: 2[5]^2-[5] = 2x25-5 = 45{1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, ...}

I

IntegerAn integer is a whole number which also includes negative numbers.{... -10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10, ...}

N

Natural

A natural number is a non negative integer:{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, ...}

NonagonalThe nonagonal number for n is given by the formula: (n(7n-5))/2 Sothe 6th nonogonal number is: ([6](7[6]-5))/2 = (6x37)/2 = 111{1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, ...}

O


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OctagonalThe octagonal number for n is given by the formula 3n2 - 2n, wheren > 0. The first few octagonal numbers are:{1, 8, 21, 40, 65, 96, 133, 176, 225, 280, ...}

OddAn odd number is an integer of the form n = 2k + 1 where k is aninteger.{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, ...}

P

PentagonalIs given by the formula: pentagonal number p = (3n^2-n)/2 for ngreater than 1. So the fifth pentagonal number is:(3[5]^2-[5])/2 = (75-5)/2 = 35{1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, ...}

PerfectThe first perfect number is 6, because 1, 2, and 3 are its properpositive divisors, and 1 + 2 + 3 = 6.{6, 28, 496, 8128, ...}

Polygonal

These are numbers that can be represented as dots that are arrangedin the shape of a regular polygon see: Triangular, Square,Pentagonal, Hexagonal, Heptagonal, Octagonal, Nonagonal, Decagonal,Dodecagonal.

PrimeA prime number (or a prime) is a natural number greater than 1 thathas no positive divisors other than 1 and itself.

Q

QuotientSee rational number.

R

RationalA rational number is a number that is represented as a quotient orfraction. Where the denominator is not zero. p/q is a fraction wherep is the numerator and q is the denominator.

S

SadA sad number or an unhappy number is a number that is not happy. seehappy numbers.

{2,3,4,5,6,7,89,11,12,14,15,16,17,18,20, ...}

Sociable1264460 = 2^2.5.17.3719, the sum of the proper divisors = 15478601547860 = 2^2.5.193.401, the sum of the proper divisors = 17276361727636 = 2^2.521.829, the sum of the proper divisors = 13051841305184 = 2^5.40787, the sum of the proper divisors = 1264460

SquareIs an integer that is a product of itself. Examples are: 1x1=1,


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2x2=4, 3x3=9, 4x4=16, 5x5=25, 6x6=36, 7x7=49, 8x8=64, ...{1,2,4,9,16,25,36,49,64,81,100,121,144, ...}

StifledA number is stifled when the digits of that number equals the baseof that number: The number 10 is stifled in base 4:2x4^1 + 2x4^0 = 10 so 22 in base 4 is equal to ten in base 10 as2+2=4.

StrangeAll strange numbers are prime. Every single digit prime number isstrange. A number with two or more digits is strange if, and onlyif, the two numbers obtained from it, by removing either its firstor its last digit, are also strange.{2, 3, 5, 7, 23, 37, 53, ...}

SurdWhen we can not simplify a number to remove a square root (or cuberoot, or nth root) then it is a surd. ie x^1/n.

T

TetrahedralA figurate number that represents a pyramid with a triangular base

and three sides, called a tetrahedron. The nth tetrahedral number isthe sum of the first n triangular numbers.{1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...}

TranscendentalAny real or complex number that is not algebraic is transcendental.Examples include e and pi.

TriangularA triangular number is the sum of n numbers starting from zero. Sosome examples are: 0, 0+1=1, 0+1+2=3, 0+1+2+3=6, 0+1+2+3+4=10, ...{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136,153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, ...}

U

UnhappyAn unhappy number or a sad number is a number that is ot happy. seehappy numbers.{2,3,4,5,6,7,89,11,12,14,15,16,17,18,20, ...}

W

WierdThe smallest weird number is 70. Its proper divisors are 1, 2, 5, 7,10, 14, and 35; these sum to 74, but no subset of these sums to 70.

{70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ...}

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Mathematics by Adi Cox