Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

AnthonyRBrown · 2024-03-09 05:08:37

Andrew Wiles: Fermat's Last theorem: abelian and non-abelian approaches

https://www.youtube.com/watch?v=4t1mgEBx1nQ

FERMATS LAST THEOREM DEMONSTRATION! BY, ANTHONY.R.BROWN (1998 SOLVED )

THE INFINITE MATHEMATICAL PATTERN OF CUBE NUMBERS!

BELOW IS MY DEMONSTRATION OF FERMATS LAST THEOREM,IT IS BASED ON THE INFINITE MATHEMATICAL PATTERN OF THE FIRST (10) CUBE NUMBERS AS A GROUP! YOU WILL SEE THAT THE LAST OR SINGLE NUMBER,FROM EACH OF THE (10) CUBE NUMBERS REPEATS ITSELF AT THE END OF EACH CUBE NUMBER, IN THE COLUMN,GIVEN IN BRACKETS () THIS PATTERN REPEATS ITSELF NO MATTER HOW LARGE OR SMALL THE CUBE NUMBERS ARE.
THE DEMONSTRATION IS! THAT BECAUSE YOU CANNOT ADD ANY TWO CUBE NUMBERS TOGETHER,TO MAKE A THIRD CUBE NUMBER,WITHIN ONE OF THE GROUPS, OR THE SAME WITHIN ANY TWO GROUPS,NO MATTER WHAT SIZE THE CUBE NUMBERS ARE! THEN YOU WILL NEVER BE ABLE TO MAKE A THIRD CUBE NUMBER,FROM ANY TWO CUBE NUMBERS.

Andrew wiles proof? (not possible! 100 %) of ( FLT ) is far too long! And contains math not really needed,with such a simple problem such as ( FLT ) plus only a few people in the world can really understand it.

Fermat stated the words below!

I HAVE DISCOVERED A TRULY MARVELLOUS DEMONSTRATION [ OF THIS GENERAL THEOREM ] WHICH THIS MARGIN IS TOO NARROW TO CONTAIN.

I believe my Demonstration must have been along the same lines as Fermats if not the same.
To understand ( FLT ) the answer is right there at the beginning, just like time itself, everything you need to know is in the first ten cube numbers!
The fact that all the numbers are used from 0 to 9 at the end of the cube numbers,the fact that some of the cubes are odd,the fact that some of the cubes are even,the fact that the cubes as a group look random!
The first ten cubes can be checked amongst themselves, for any two cubes that may make a third,and checked with any other ten as part of their group.
The infinite unique pattern is what makes everything work! Once you understand how cubes are made,then you have found all the cubes!
p.'s except the last!! The last case is unsolvable! for this reason its impossible to have a 100 % Proof! ( and applies to any infinite math problem ) only a Demonstration,as Fermat himself said!

BELOW THE FIRST TEN CUBE NUMBERS.

CUBE = (1): PATTERN = (1): COUNT = 1
CUBE = (8): PATTERN = (8): COUNT = 2
CUBE = 2(7): PATTERN = 2(7): COUNT = 3
CUBE = 6(4): PATTERN = 6(4): COUNT = 4
CUBE = 12(5): PATTERN = 12(5): COUNT = 5
CUBE = 21(6): PATTERN = 21(6): COUNT = 6
CUBE = 34(3): PATTERN = 34(3): COUNT = 7
CUBE = 51(2): PATTERN = 51(2): COUNT = 8
CUBE = 72(9): PATTERN = 72(9): COUNT = 9
CUBE = 100(0): PATTERN = (0): COUNT = 10

BELOW THE NEXT TEN.

CUBE = 133(1): PATTERN = (1): COUNT = 11
CUBE = 172(8): PATTERN = (8): COUNT = 12
CUBE = 219(7): PATTERN = 2(7): COUNT = 13
CUBE = 274(4): PATTERN = 6(4): COUNT = 14
CUBE = 337(5): PATTERN = 12(5): COUNT = 15
CUBE = 409(6): PATTERN = 21(6): COUNT = 16
CUBE = 491(3): PATTERN = 34(3): COUNT = 17
CUBE = 583(2): PATTERN = 51(2): COUNT = 18
CUBE = 685(9): PATTERN = 72(9): COUNT = 19
CUBE = 800(0): PATTERN = (0): COUNT = 20

AND THE NEXT TEN ETC.!

CUBE = 926(1): PATTERN = (1): COUNT = 21
CUBE = 1064(8): PATTERN = (8): COUNT = 22
CUBE = 1216(7): PATTERN = 2(7): COUNT = 23
CUBE = 1382(4): PATTERN = 6(4): COUNT = 24
CUBE = 1562(5): PATTERN = 12(5): COUNT = 25
CUBE = 1757(6): PATTERN = 21(6): COUNT = 26
CUBE = 1968(3): PATTERN = 34(3): COUNT = 27
CUBE = 2195(2): PATTERN = 51(2): COUNT = 28
CUBE = 2438(9): PATTERN = 72(9): COUNT = 29
CUBE = 2700(0): PATTERN = (0): COUNT = 30

EXAMPLE BELOW : TRY TO FIND IF THE TWO NUMBERS BELOW ADD TOGETHER MAKE A CUBE NUMBER!

2685619 + 4173281 = 6858900

Following what I have said in my Demonstration,all cube numbers are made in the same way,the infinite pattern is unique in the fact that the numbers 0 to 9 are only used once as in PATTERN A, at the end of the cubes numbers,within each group of any ten! plus more proof further on.
What it is like is a Mathematical engine that produces the cube numbers in unique groups of ten,starting from the first group of ten onwards Infinitely.
The sum of the two numbers above ( 6858900 ) is not a cube number! the reason its not a cube number is because it does not fit in any of the groups of ten,that are produced!
The nearest cube number too ( 6858900 ) is ( 6859000 ) this cube number is part of the 19 th Group given below,and is just before the start of the 20 th Group.
I will explain more how the Demonstration works below!

19 th Group

CUBE = 5929741 : PATTERN A = (1)
CUBE = 6028568 : PATTERN A = (8)
CUBE = 6128487 : PATTERN A = 2(7)
CUBE = 6229504 : PATTERN A = 6(4)
CUBE = 6331625 : PATTERN A = 12(5)
CUBE = 6434856 : PATTERN A = 21(6)
CUBE = 6539203 : PATTERN A = 34(3)
CUBE = 6644672 : PATTERN A = 51(2)
CUBE = 6751269 : PATTERN A = 72(9)
CUBE = 6859000 : PATTERN A = (0)

20 th Group

CUBE = 6967871 : PATTERN A = (1)
CUBE = 7077888 : PATTERN A = (8)
CUBE = 7189057 : PATTERN A = 2(7)
CUBE = 7301384 : PATTERN A = 6(4)
CUBE = 7414875 : PATTERN A = 12(5)
CUBE = 7529536 : PATTERN A = 21(6)
CUBE = 7645373 : PATTERN A = 34(3)
CUBE = 7762392 : PATTERN A = 51(2)
CUBE = 7880599 : PATTERN A = 72(9)
CUBE = 8000000 : PATTERN A = (0)

MORE INFORMATION BELOW ON HOW MY DEMONSTRATION WORKS!

The next pattern is the group of numbers that are produced, by taking away the cube numbers starting with the largest and working backwards,as in the first group below,so 1000 - 729 = (271),729 - 512 = (217),512 - 343 = (169),343 - 216 = (127),216 -125 = (91),125 - 64 = (61),64 - 27 = (37),27 - 8 = (19),8 - 1 = (7).
What we now have is the second unique pattern (B),that shows us we have a group of related cube numbers,the end numbers are made up of 7,9,7,1,1,7,9,7,1 in brackets ()

1 st Group PATTERN B

CUBE = 1
DIFFERENCE = 7 : PATTERN B = (7)
CUBE = 8
DIFFERENCE = 19 : PATTERN B = (9)
CUBE = 27
DIFFERENCE = 37 : PATTERN B = (7)
CUBE = 64
DIFFERENCE = 61 : PATTERN B = (1)
CUBE = 125
DIFFERENCE = 91 : PATTERN B = (1)
CUBE = 216
DIFFERENCE = 127 : PATTERN B = (7)
CUBE = 343
DIFFERENCE = 169 : PATTERN B = (9)
CUBE = 512
DIFFERENCE = 217 : PATTERN B = (7)
CUBE = 729
DIFFERENCE = 271 : PATTERN B = (1)
CUBE = 1000

Now we move on to pattern (C) this pattern is made by dividing the actual COUNT position,in relation to where the cube numbers are from the start onwards! NOTE: the resulting numbers are always WHOLE numbers! the end numbers are made up of 1,4,9,6,5,6,9,4,1,0 in brackets ()

1 st Group PATTERN C

CUBE = 1 : COUNT = 1 : EQUALS 1 : PATTERN C = (1)
CUBE = 8 : COUNT = 2 : EQUALS 4 : PATTERN C = (4)
CUBE = 27 : COUNT = 3 : EQUALS 9 : PATTERN C = (9)
CUBE = 64 : COUNT = 4 : EQUALS 16 : PATTERN C = (6)
CUBE = 125 : COUNT = 5 : EQUALS 25 : PATTERN C = (5)
CUBE = 216 : COUNT = 6 : EQUALS 36 : PATTERN C = (6)
CUBE = 343 : COUNT = 7 : EQUALS 49 : PATTERN C = (9)
CUBE = 512 : COUNT = 8 : EQUALS 64 : PATTERN C = (4)
CUBE = 729 : COUNT = 9 : EQUALS 81 : PATTERN C = (1)
CUBE = 1000 : COUNT = 10 : EQUALS 100 : PATTERN C = (0)

Now lets see if the number ( 6858900 ) the sum of the two numbers is a cube number or not! I will put the number in the group where the number is closest matched,which happens to be the tenth place within the 19 th Group below,and replace the true cube number ( 6859000 ) with the number ( 6858900 ) and see what happens to the patterns A,B,C!

19 th Group

CUBE = 5929741 : PATTERN A = (1) : COUNT = 181 : EQUALS 32761 : PATTERN C = (1)
DIFFERENCE = 98827 : PATTERN B = (7)
CUBE = 6028568 : PATTERN A = (8) : COUNT = 182 : EQUALS 33124 : PATTERN C = (4)
DIFFERENCE = 99919 : PATTERN B = (9)
CUBE = 6128487 : PATTERN A = 2(7) : COUNT = 183 : EQUALS 33489 : PATTERN C = (9)
DIFFERENCE = 101017 : PATTERN B = (7)
CUBE = 6229504 : PATTERN A = 6(4) : COUNT = 184 : EQUALS 33856 : PATTERN C = (6)
DIFFERENCE = 102121 : PATTERN B = (1)
CUBE = 6331625 : PATTERN A = 12(5) : COUNT = 185 : EQUALS 34225 : PATTERN C = (5)
DIFFERENCE = 103231 : PATTERN B = (1)
CUBE = 6434856 : PATTERN A = 21(6) : COUNT = 186 : EQUALS 34596 : PATTERN C = (6)
DIFFERENCE = 104347 : PATTERN B = (7)
CUBE = 6539203 : PATTERN A = 34(3) : COUNT = 187 : EQUALS 34969 : PATTERN C = (9)
DIFFERENCE = 105469 : PATTERN B = (9)
CUBE = 6644672 : PATTERN A = 51(2) : COUNT = 188 : EQUALS 35344 : PATTERN C = (4)
DIFFERENCE = 106597 : PATTERN B = (7)
CUBE = 6751269 : PATTERN A = 72(9) : COUNT = 189 : EQUALS 35721 : PATTERN C = (1)

Below is the tenth place!

Below is the true cube number,and shows all three Patterns working as normal!

DIFFERENCE = 107731 : PATTERN B = (1)
CUBE = 6859000 : PATTERN A = (0) : COUNT = 190 : EQUALS 36100 : PATTERN C = (0)

Below is the new number,you will notice the number works with only two of the patterns,but is way out with the third,i.e. not a WHOLE number! and ending with the wrong number in the () this proves its not a cube number!

DIFFERENCE = 107631 : PATTERN B = (1)
CUBE = 6858900 : PATTERN A = (0) : COUNT = 190 : EQUALS 36099.473 : PATTERN C = (int(9)? or (3)? both are wrong!

MORE INFORMATION BELOW ON HOW MY DEMONSTRATION WORKS!

As I have already said,all cube numbers are made the same way using what I would like to call a MATHEMATICAL ENGINE!
the process is from the first single cube number (1) onwards.
The most important numbers are the end numbers in brackets () the reason they are the most important is because they are the end of the process for making cube numbers! they are if you like the TRUTH numbers! its a bit like the numbers go round and round mathematically,and eventually make a whole cube number!
I have put forward three patterns A,B,C that show how these TRUTH numbers are present in any group of ten cube numbers,I am again giving the 1st and 19th Groups as examples!
The size of the cube numbers never changes how these TRUTH numbers are made! another way of looking at what im saying is that the cube numbers in the 1st Group are the same as in the 19th Group by the way they are made, and both are the same as any other group! size does not affect the TRUTH number results!
So if these two groups are the same,then trying to find if any two cube numbers make a third ,within the first group! is the same as trying with any other group,then after that test! we only need to test two groups against each other,because the maximum test is two from one group,or one from each group!
Now if we want to find out if a group of any ten numbers are cube numbers,the ten numbers must pass the three tests,they must conform to the patterns A,B,C and at the same time if we want to test if any single number is a cube number,this can be done by finding if it fits in with the nine cube numbers that are related to it,as part of its own group!.

PATTERN (A) = 1,8,7,4,5,6,3,2,9,0 in brackets ()

PATTERN (B) = 7,9,7,1,1,7,9,7,1 in brackets ()

PATTERN (C) = 1,4,9,6,5,6,9,4,1,0 in brackets ()

1 st Group

CUBE = 1: PATTERN A = (1) : COUNT = 1 : EQUALS 1 : PATTERN C = (1)
DIFFERENCE = 7 : PATTERN B = (7)
CUBE = 8: PATTERN A = (8) : COUNT = 2 : EQUALS 4 : PATTERN C = (4)
DIFFERENCE = 19 : PATTERN B = (9)
CUBE = 27: PATTERN A = (7) : COUNT = 3 : EQUALS 9 : PATTERN C = (9)
DIFFERENCE = 37 : PATTERN B = (7)
CUBE = 64: PATTERN A = (4) : COUNT = 4 : EQUALS 16 : PATTERN C = (6)
DIFFERENCE = 61 : PATTERN B = (1)
CUBE = 125: PATTERN A = (5) : COUNT = 5 : EQUALS 25 : PATTERN C = (5)
DIFFERENCE = 91 : PATTERN B = (1)
CUBE = 216: PATTERN A = (6) : COUNT = 6 : EQUALS 36 : PATTERN C = (6)
DIFFERENCE = 127 : PATTERN B = (7)
CUBE = 343: PATTERN A = (3) : COUNT = 7 : EQUALS 49 : PATTERN C = (9)
DIFFERENCE = 169 : PATTERN B = (9)
CUBE = 512: PATTERN A = (2) : COUNT = 8 : EQUALS 64 : PATTERN C = (4)
DIFFERENCE = 217 : PATTERN B = (7)
CUBE = 729: PATTERN A = (9) : COUNT = 9 : EQUALS 81 : PATTERN C = (1)
DIFFERENCE = 271 : PATTERN B = (1)
CUBE = 1000: PATTERN A = (0) : COUNT = 10 : EQUALS 100 : PATTERN C = (0)

19 th Group

CUBE = 5929741 : PATTERN A = (1) : COUNT = 181 : EQUALS 32761 : PATTERN C = (1)
DIFFERENCE = 98827 : PATTERN B = (7)
CUBE = 6028568 : PATTERN A = (8) : COUNT = 182 : EQUALS 33124 : PATTERN C = (4)
DIFFERENCE = 99919 : PATTERN B = (9)
CUBE = 6128487 : PATTERN A = (7) : COUNT = 183 : EQUALS 33489 : PATTERN C = (9)
DIFFERENCE = 101017 : PATTERN B = (7)
CUBE = 6229504 : PATTERN A = (4) : COUNT = 184 : EQUALS 33856 : PATTERN C = (6)
DIFFERENCE = 102121 : PATTERN B = (1)
CUBE = 6331625 : PATTERN A = (5) : COUNT = 185 : EQUALS 34225 : PATTERN C = (5)
DIFFERENCE = 103231 : PATTERN B = (1)
CUBE = 6434856 : PATTERN A = (6) : COUNT = 186 : EQUALS 34596 : PATTERN C = (6)
DIFFERENCE = 104347 : PATTERN B = (7)
CUBE = 6539203 : PATTERN A = (3) : COUNT = 187 : EQUALS 34969 : PATTERN C = (9)
DIFFERENCE = 105469 : PATTERN B = (9)
CUBE = 6644672 : PATTERN A = (2) : COUNT = 188 : EQUALS 35344 : PATTERN C = (4)
DIFFERENCE = 106597 : PATTERN B = (7)
CUBE = 6751269 : PATTERN A = (9) : COUNT = 189 : EQUALS 35721 : PATTERN C = (1)
DIFFERENCE = 107731 : PATTERN B = (1)
CUBE = 6859000 : PATTERN A = (0) : COUNT = 190 : EQUALS 36100 : PATTERN C = (0)

THE ABOVE EXAMPLES CLEARLY SHOW ALL CUBE NUMBERS ARE MADE THE SAME WAY! SO THE METHODS USED TO SEE IF ANY TWO NUMBERS MAKE A CUBE NUMBER, WILL ALWAYS BE THE SAME! REGARDLESS OF THE SIZE OF THE NUMBERS.

Please forward any comments to me at my E-MAIL address below,or to any Forum this (FLT) Demonstration is on!

[email protected]

A.R.B

Last edited by AnthonyRBrown (2024-04-28 21:58:09)

AnthonyRBrown · 2024-03-09 05:10:33

Below is a link to my FERMATS LAST THEOREM DEMONSTRATION PROGRAM 1998 (Original)

github.com/TARBCS/The-A.R.B-Strongest-Games-Programs-In-The-World

Below shows the 1st 1000 Cubes produced by my MAKE CUBE NUMBER PROGRAM!

CUBE = 1 : PATTERN = 1 : COUNT = 1
CUBE = 8 : PATTERN = 8 : COUNT = 2
CUBE = 27 : PATTERN = 27 : COUNT = 3
CUBE = 64 : PATTERN = 64 : COUNT = 4
CUBE = 125 : PATTERN = 125 : COUNT = 5
CUBE = 216 : PATTERN = 216 : COUNT = 6
CUBE = 343 : PATTERN = 343 : COUNT = 7
CUBE = 512 : PATTERN = 512 : COUNT = 8
CUBE = 729 : PATTERN = 729 : COUNT = 9
CUBE = 1000 : PATTERN = 0 : COUNT = 10
CUBE = 1331 : PATTERN = 1 : COUNT = 11
CUBE = 1728 : PATTERN = 8 : COUNT = 12
CUBE = 2197 : PATTERN = 27 : COUNT = 13
CUBE = 2744 : PATTERN = 64 : COUNT = 14
CUBE = 3375 : PATTERN = 125 : COUNT = 15
CUBE = 4096 : PATTERN = 216 : COUNT = 16
CUBE = 4913 : PATTERN = 343 : COUNT = 17
CUBE = 5832 : PATTERN = 512 : COUNT = 18
CUBE = 6859 : PATTERN = 729 : COUNT = 19
CUBE = 8000 : PATTERN = 0 : COUNT = 20
CUBE = 9261 : PATTERN = 1 : COUNT = 21
CUBE = 10648 : PATTERN = 8 : COUNT = 22
CUBE = 12167 : PATTERN = 27 : COUNT = 23
CUBE = 13824 : PATTERN = 64 : COUNT = 24
CUBE = 15625 : PATTERN = 125 : COUNT = 25
CUBE = 17576 : PATTERN = 216 : COUNT = 26
CUBE = 19683 : PATTERN = 343 : COUNT = 27
CUBE = 21952 : PATTERN = 512 : COUNT = 28
CUBE = 24389 : PATTERN = 729 : COUNT = 29
CUBE = 27000 : PATTERN = 0 : COUNT = 30
CUBE = 29791 : PATTERN = 1 : COUNT = 31
CUBE = 32768 : PATTERN = 8 : COUNT = 32
CUBE = 35937 : PATTERN = 27 : COUNT = 33
CUBE = 39304 : PATTERN = 64 : COUNT = 34
CUBE = 42875 : PATTERN = 125 : COUNT = 35
CUBE = 46656 : PATTERN = 216 : COUNT = 36
CUBE = 50653 : PATTERN = 343 : COUNT = 37
CUBE = 54872 : PATTERN = 512 : COUNT = 38
CUBE = 59319 : PATTERN = 729 : COUNT = 39
CUBE = 64000 : PATTERN = 0 : COUNT = 40
CUBE = 68921 : PATTERN = 1 : COUNT = 41
CUBE = 74088 : PATTERN = 8 : COUNT = 42
CUBE = 79507 : PATTERN = 27 : COUNT = 43
CUBE = 85184 : PATTERN = 64 : COUNT = 44
CUBE = 91125 : PATTERN = 125 : COUNT = 45
CUBE = 97336 : PATTERN = 216 : COUNT = 46
CUBE = 103823 : PATTERN = 343 : COUNT = 47
CUBE = 110592 : PATTERN = 512 : COUNT = 48
CUBE = 117649 : PATTERN = 729 : COUNT = 49
CUBE = 125000 : PATTERN = 0 : COUNT = 50
CUBE = 132651 : PATTERN = 1 : COUNT = 51
CUBE = 140608 : PATTERN = 8 : COUNT = 52
CUBE = 148877 : PATTERN = 27 : COUNT = 53
CUBE = 157464 : PATTERN = 64 : COUNT = 54
CUBE = 166375 : PATTERN = 125 : COUNT = 55
CUBE = 175616 : PATTERN = 216 : COUNT = 56
CUBE = 185193 : PATTERN = 343 : COUNT = 57
CUBE = 195112 : PATTERN = 512 : COUNT = 58
CUBE = 205379 : PATTERN = 729 : COUNT = 59
CUBE = 216000 : PATTERN = 0 : COUNT = 60
CUBE = 226981 : PATTERN = 1 : COUNT = 61
CUBE = 238328 : PATTERN = 8 : COUNT = 62
CUBE = 250047 : PATTERN = 27 : COUNT = 63
CUBE = 262144 : PATTERN = 64 : COUNT = 64
CUBE = 274625 : PATTERN = 125 : COUNT = 65
CUBE = 287496 : PATTERN = 216 : COUNT = 66
CUBE = 300763 : PATTERN = 343 : COUNT = 67
CUBE = 314432 : PATTERN = 512 : COUNT = 68
CUBE = 328509 : PATTERN = 729 : COUNT = 69
CUBE = 343000 : PATTERN = 0 : COUNT = 70
CUBE = 357911 : PATTERN = 1 : COUNT = 71
CUBE = 373248 : PATTERN = 8 : COUNT = 72
CUBE = 389017 : PATTERN = 27 : COUNT = 73
CUBE = 405224 : PATTERN = 64 : COUNT = 74
CUBE = 421875 : PATTERN = 125 : COUNT = 75
CUBE = 438976 : PATTERN = 216 : COUNT = 76
CUBE = 456533 : PATTERN = 343 : COUNT = 77
CUBE = 474552 : PATTERN = 512 : COUNT = 78
CUBE = 493039 : PATTERN = 729 : COUNT = 79
CUBE = 512000 : PATTERN = 0 : COUNT = 80
CUBE = 531441 : PATTERN = 1 : COUNT = 81
CUBE = 551368 : PATTERN = 8 : COUNT = 82
CUBE = 571787 : PATTERN = 27 : COUNT = 83
CUBE = 592704 : PATTERN = 64 : COUNT = 84
CUBE = 614125 : PATTERN = 125 : COUNT = 85
CUBE = 636056 : PATTERN = 216 : COUNT = 86
CUBE = 658503 : PATTERN = 343 : COUNT = 87
CUBE = 681472 : PATTERN = 512 : COUNT = 88
CUBE = 704969 : PATTERN = 729 : COUNT = 89
CUBE = 729000 : PATTERN = 0 : COUNT = 90
CUBE = 753571 : PATTERN = 1 : COUNT = 91
CUBE = 778688 : PATTERN = 8 : COUNT = 92
CUBE = 804357 : PATTERN = 27 : COUNT = 93
CUBE = 830584 : PATTERN = 64 : COUNT = 94
CUBE = 857375 : PATTERN = 125 : COUNT = 95
CUBE = 884736 : PATTERN = 216 : COUNT = 96
CUBE = 912673 : PATTERN = 343 : COUNT = 97
CUBE = 941192 : PATTERN = 512 : COUNT = 98
CUBE = 970299 : PATTERN = 729 : COUNT = 99
CUBE = 1000000 : PATTERN = 0 : COUNT = 100
CUBE = 1030301 : PATTERN = 1 : COUNT = 101
CUBE = 1061208 : PATTERN = 8 : COUNT = 102
CUBE = 1092727 : PATTERN = 27 : COUNT = 103
CUBE = 1124864 : PATTERN = 64 : COUNT = 104
CUBE = 1157625 : PATTERN = 125 : COUNT = 105
CUBE = 1191016 : PATTERN = 216 : COUNT = 106
CUBE = 1225043 : PATTERN = 343 : COUNT = 107
CUBE = 1259712 : PATTERN = 512 : COUNT = 108
CUBE = 1295029 : PATTERN = 729 : COUNT = 109
CUBE = 1331000 : PATTERN = 0 : COUNT = 110
CUBE = 1367631 : PATTERN = 1 : COUNT = 111
CUBE = 1404928 : PATTERN = 8 : COUNT = 112
CUBE = 1442897 : PATTERN = 27 : COUNT = 113
CUBE = 1481544 : PATTERN = 64 : COUNT = 114
CUBE = 1520875 : PATTERN = 125 : COUNT = 115
CUBE = 1560896 : PATTERN = 216 : COUNT = 116
CUBE = 1601613 : PATTERN = 343 : COUNT = 117
CUBE = 1643032 : PATTERN = 512 : COUNT = 118
CUBE = 1685159 : PATTERN = 729 : COUNT = 119
CUBE = 1728000 : PATTERN = 0 : COUNT = 120
CUBE = 1771561 : PATTERN = 1 : COUNT = 121
CUBE = 1815848 : PATTERN = 8 : COUNT = 122
CUBE = 1860867 : PATTERN = 27 : COUNT = 123
CUBE = 1906624 : PATTERN = 64 : COUNT = 124
CUBE = 1953125 : PATTERN = 125 : COUNT = 125
CUBE = 2000376 : PATTERN = 216 : COUNT = 126
CUBE = 2048383 : PATTERN = 343 : COUNT = 127
CUBE = 2097152 : PATTERN = 512 : COUNT = 128
CUBE = 2146689 : PATTERN = 729 : COUNT = 129
CUBE = 2197000 : PATTERN = 0 : COUNT = 130
CUBE = 2248091 : PATTERN = 1 : COUNT = 131
CUBE = 2299968 : PATTERN = 8 : COUNT = 132
CUBE = 2352637 : PATTERN = 27 : COUNT = 133
CUBE = 2406104 : PATTERN = 64 : COUNT = 134
CUBE = 2460375 : PATTERN = 125 : COUNT = 135
CUBE = 2515456 : PATTERN = 216 : COUNT = 136
CUBE = 2571353 : PATTERN = 343 : COUNT = 137
CUBE = 2628072 : PATTERN = 512 : COUNT = 138
CUBE = 2685619 : PATTERN = 729 : COUNT = 139
CUBE = 2744000 : PATTERN = 0 : COUNT = 140
CUBE = 2803221 : PATTERN = 1 : COUNT = 141
CUBE = 2863288 : PATTERN = 8 : COUNT = 142
CUBE = 2924207 : PATTERN = 27 : COUNT = 143
CUBE = 2985984 : PATTERN = 64 : COUNT = 144
CUBE = 3048625 : PATTERN = 125 : COUNT = 145
CUBE = 3112136 : PATTERN = 216 : COUNT = 146
CUBE = 3176523 : PATTERN = 343 : COUNT = 147
CUBE = 3241792 : PATTERN = 512 : COUNT = 148
CUBE = 3307949 : PATTERN = 729 : COUNT = 149
CUBE = 3375000 : PATTERN = 0 : COUNT = 150
CUBE = 3442951 : PATTERN = 1 : COUNT = 151
CUBE = 3511808 : PATTERN = 8 : COUNT = 152
CUBE = 3581577 : PATTERN = 27 : COUNT = 153
CUBE = 3652264 : PATTERN = 64 : COUNT = 154
CUBE = 3723875 : PATTERN = 125 : COUNT = 155
CUBE = 3796416 : PATTERN = 216 : COUNT = 156
CUBE = 3869893 : PATTERN = 343 : COUNT = 157
CUBE = 3944312 : PATTERN = 512 : COUNT = 158
CUBE = 4019679 : PATTERN = 729 : COUNT = 159
CUBE = 4096000 : PATTERN = 0 : COUNT = 160
CUBE = 4173281 : PATTERN = 1 : COUNT = 161
CUBE = 4251528 : PATTERN = 8 : COUNT = 162
CUBE = 4330747 : PATTERN = 27 : COUNT = 163
CUBE = 4410944 : PATTERN = 64 : COUNT = 164
CUBE = 4492125 : PATTERN = 125 : COUNT = 165
CUBE = 4574296 : PATTERN = 216 : COUNT = 166
CUBE = 4657463 : PATTERN = 343 : COUNT = 167
CUBE = 4741632 : PATTERN = 512 : COUNT = 168
CUBE = 4826809 : PATTERN = 729 : COUNT = 169
CUBE = 4913000 : PATTERN = 0 : COUNT = 170
CUBE = 5000211 : PATTERN = 1 : COUNT = 171
CUBE = 5088448 : PATTERN = 8 : COUNT = 172
CUBE = 5177717 : PATTERN = 27 : COUNT = 173
CUBE = 5268024 : PATTERN = 64 : COUNT = 174
CUBE = 5359375 : PATTERN = 125 : COUNT = 175
CUBE = 5451776 : PATTERN = 216 : COUNT = 176
CUBE = 5545233 : PATTERN = 343 : COUNT = 177
CUBE = 5639752 : PATTERN = 512 : COUNT = 178
CUBE = 5735339 : PATTERN = 729 : COUNT = 179
CUBE = 5832000 : PATTERN = 0 : COUNT = 180
CUBE = 5929741 : PATTERN = 1 : COUNT = 181
CUBE = 6028568 : PATTERN = 8 : COUNT = 182
CUBE = 6128487 : PATTERN = 27 : COUNT = 183
CUBE = 6229504 : PATTERN = 64 : COUNT = 184
CUBE = 6331625 : PATTERN = 125 : COUNT = 185
CUBE = 6434856 : PATTERN = 216 : COUNT = 186
CUBE = 6539203 : PATTERN = 343 : COUNT = 187
CUBE = 6644672 : PATTERN = 512 : COUNT = 188
CUBE = 6751269 : PATTERN = 729 : COUNT = 189
CUBE = 6859000 : PATTERN = 0 : COUNT = 190
CUBE = 6967871 : PATTERN = 1 : COUNT = 191
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CUBE = 275894451 : PATTERN = 1 : COUNT = 651
CUBE = 277167808 : PATTERN = 8 : COUNT = 652
CUBE = 278445077 : PATTERN = 27 : COUNT = 653
CUBE = 279726264 : PATTERN = 64 : COUNT = 654
CUBE = 281011375 : PATTERN = 125 : COUNT = 655
CUBE = 282300416 : PATTERN = 216 : COUNT = 656
CUBE = 283593393 : PATTERN = 343 : COUNT = 657
CUBE = 284890312 : PATTERN = 512 : COUNT = 658
CUBE = 286191179 : PATTERN = 729 : COUNT = 659
CUBE = 287496000 : PATTERN = 0 : COUNT = 660
CUBE = 288804781 : PATTERN = 1 : COUNT = 661
CUBE = 290117528 : PATTERN = 8 : COUNT = 662
CUBE = 291434247 : PATTERN = 27 : COUNT = 663
CUBE = 292754944 : PATTERN = 64 : COUNT = 664
CUBE = 294079625 : PATTERN = 125 : COUNT = 665
CUBE = 295408296 : PATTERN = 216 : COUNT = 666
CUBE = 296740963 : PATTERN = 343 : COUNT = 667
CUBE = 298077632 : PATTERN = 512 : COUNT = 668
CUBE = 299418309 : PATTERN = 729 : COUNT = 669
CUBE = 300763000 : PATTERN = 0 : COUNT = 670
CUBE = 302111711 : PATTERN = 1 : COUNT = 671
CUBE = 303464448 : PATTERN = 8 : COUNT = 672
CUBE = 304821217 : PATTERN = 27 : COUNT = 673
CUBE = 306182024 : PATTERN = 64 : COUNT = 674
CUBE = 307546875 : PATTERN = 125 : COUNT = 675
CUBE = 308915776 : PATTERN = 216 : COUNT = 676
CUBE = 310288733 : PATTERN = 343 : COUNT = 677
CUBE = 311665752 : PATTERN = 512 : COUNT = 678
CUBE = 313046839 : PATTERN = 729 : COUNT = 679
CUBE = 314432000 : PATTERN = 0 : COUNT = 680
CUBE = 315821241 : PATTERN = 1 : COUNT = 681
CUBE = 317214568 : PATTERN = 8 : COUNT = 682
CUBE = 318611987 : PATTERN = 27 : COUNT = 683
CUBE = 320013504 : PATTERN = 64 : COUNT = 684
CUBE = 321419125 : PATTERN = 125 : COUNT = 685
CUBE = 322828856 : PATTERN = 216 : COUNT = 686
CUBE = 324242703 : PATTERN = 343 : COUNT = 687
CUBE = 325660672 : PATTERN = 512 : COUNT = 688
CUBE = 327082769 : PATTERN = 729 : COUNT = 689
CUBE = 328509000 : PATTERN = 0 : COUNT = 690
CUBE = 329939371 : PATTERN = 1 : COUNT = 691
CUBE = 331373888 : PATTERN = 8 : COUNT = 692
CUBE = 332812557 : PATTERN = 27 : COUNT = 693
CUBE = 334255384 : PATTERN = 64 : COUNT = 694
CUBE = 335702375 : PATTERN = 125 : COUNT = 695
CUBE = 337153536 : PATTERN = 216 : COUNT = 696
CUBE = 338608873 : PATTERN = 343 : COUNT = 697
CUBE = 340068392 : PATTERN = 512 : COUNT = 698
CUBE = 341532099 : PATTERN = 729 : COUNT = 699
CUBE = 343000000 : PATTERN = 0 : COUNT = 700
CUBE = 344472101 : PATTERN = 1 : COUNT = 701
CUBE = 345948408 : PATTERN = 8 : COUNT = 702
CUBE = 347428927 : PATTERN = 27 : COUNT = 703
CUBE = 348913664 : PATTERN = 64 : COUNT = 704
CUBE = 350402625 : PATTERN = 125 : COUNT = 705
CUBE = 351895816 : PATTERN = 216 : COUNT = 706
CUBE = 353393243 : PATTERN = 343 : COUNT = 707
CUBE = 354894912 : PATTERN = 512 : COUNT = 708
CUBE = 356400829 : PATTERN = 729 : COUNT = 709
CUBE = 357911000 : PATTERN = 0 : COUNT = 710
CUBE = 359425431 : PATTERN = 1 : COUNT = 711
CUBE = 360944128 : PATTERN = 8 : COUNT = 712
CUBE = 362467097 : PATTERN = 27 : COUNT = 713
CUBE = 363994344 : PATTERN = 64 : COUNT = 714
CUBE = 365525875 : PATTERN = 125 : COUNT = 715
CUBE = 367061696 : PATTERN = 216 : COUNT = 716
CUBE = 368601813 : PATTERN = 343 : COUNT = 717
CUBE = 370146232 : PATTERN = 512 : COUNT = 718
CUBE = 371694959 : PATTERN = 729 : COUNT = 719
CUBE = 373248000 : PATTERN = 0 : COUNT = 720
CUBE = 374805361 : PATTERN = 1 : COUNT = 721
CUBE = 376367048 : PATTERN = 8 : COUNT = 722
CUBE = 377933067 : PATTERN = 27 : COUNT = 723
CUBE = 379503424 : PATTERN = 64 : COUNT = 724
CUBE = 381078125 : PATTERN = 125 : COUNT = 725
CUBE = 382657176 : PATTERN = 216 : COUNT = 726
CUBE = 384240583 : PATTERN = 343 : COUNT = 727
CUBE = 385828352 : PATTERN = 512 : COUNT = 728
CUBE = 387420489 : PATTERN = 729 : COUNT = 729
CUBE = 389017000 : PATTERN = 0 : COUNT = 730
CUBE = 390617891 : PATTERN = 1 : COUNT = 731
CUBE = 392223168 : PATTERN = 8 : COUNT = 732
CUBE = 393832837 : PATTERN = 27 : COUNT = 733
CUBE = 395446904 : PATTERN = 64 : COUNT = 734
CUBE = 397065375 : PATTERN = 125 : COUNT = 735
CUBE = 398688256 : PATTERN = 216 : COUNT = 736
CUBE = 400315553 : PATTERN = 343 : COUNT = 737
CUBE = 401947272 : PATTERN = 512 : COUNT = 738
CUBE = 403583419 : PATTERN = 729 : COUNT = 739
CUBE = 405224000 : PATTERN = 0 : COUNT = 740
CUBE = 406869021 : PATTERN = 1 : COUNT = 741
CUBE = 408518488 : PATTERN = 8 : COUNT = 742
CUBE = 410172407 : PATTERN = 27 : COUNT = 743
CUBE = 411830784 : PATTERN = 64 : COUNT = 744
CUBE = 413493625 : PATTERN = 125 : COUNT = 745
CUBE = 415160936 : PATTERN = 216 : COUNT = 746
CUBE = 416832723 : PATTERN = 343 : COUNT = 747
CUBE = 418508992 : PATTERN = 512 : COUNT = 748
CUBE = 420189749 : PATTERN = 729 : COUNT = 749
CUBE = 421875000 : PATTERN = 0 : COUNT = 750
CUBE = 423564751 : PATTERN = 1 : COUNT = 751
CUBE = 425259008 : PATTERN = 8 : COUNT = 752
CUBE = 426957777 : PATTERN = 27 : COUNT = 753
CUBE = 428661064 : PATTERN = 64 : COUNT = 754
CUBE = 430368875 : PATTERN = 125 : COUNT = 755
CUBE = 432081216 : PATTERN = 216 : COUNT = 756
CUBE = 433798093 : PATTERN = 343 : COUNT = 757
CUBE = 435519512 : PATTERN = 512 : COUNT = 758
CUBE = 437245479 : PATTERN = 729 : COUNT = 759
CUBE = 438976000 : PATTERN = 0 : COUNT = 760
CUBE = 440711081 : PATTERN = 1 : COUNT = 761
CUBE = 442450728 : PATTERN = 8 : COUNT = 762
CUBE = 444194947 : PATTERN = 27 : COUNT = 763
CUBE = 445943744 : PATTERN = 64 : COUNT = 764
CUBE = 447697125 : PATTERN = 125 : COUNT = 765
CUBE = 449455096 : PATTERN = 216 : COUNT = 766
CUBE = 451217663 : PATTERN = 343 : COUNT = 767
CUBE = 452984832 : PATTERN = 512 : COUNT = 768
CUBE = 454756609 : PATTERN = 729 : COUNT = 769
CUBE = 456533000 : PATTERN = 0 : COUNT = 770
CUBE = 458314011 : PATTERN = 1 : COUNT = 771
CUBE = 460099648 : PATTERN = 8 : COUNT = 772
CUBE = 461889917 : PATTERN = 27 : COUNT = 773
CUBE = 463684824 : PATTERN = 64 : COUNT = 774
CUBE = 465484375 : PATTERN = 125 : COUNT = 775
CUBE = 467288576 : PATTERN = 216 : COUNT = 776
CUBE = 469097433 : PATTERN = 343 : COUNT = 777
CUBE = 470910952 : PATTERN = 512 : COUNT = 778
CUBE = 472729139 : PATTERN = 729 : COUNT = 779
CUBE = 474552000 : PATTERN = 0 : COUNT = 780
CUBE = 476379541 : PATTERN = 1 : COUNT = 781
CUBE = 478211768 : PATTERN = 8 : COUNT = 782
CUBE = 480048687 : PATTERN = 27 : COUNT = 783
CUBE = 481890304 : PATTERN = 64 : COUNT = 784
CUBE = 483736625 : PATTERN = 125 : COUNT = 785
CUBE = 485587656 : PATTERN = 216 : COUNT = 786
CUBE = 487443403 : PATTERN = 343 : COUNT = 787
CUBE = 489303872 : PATTERN = 512 : COUNT = 788
CUBE = 491169069 : PATTERN = 729 : COUNT = 789
CUBE = 493039000 : PATTERN = 0 : COUNT = 790
CUBE = 494913671 : PATTERN = 1 : COUNT = 791
CUBE = 496793088 : PATTERN = 8 : COUNT = 792
CUBE = 498677257 : PATTERN = 27 : COUNT = 793
CUBE = 500566184 : PATTERN = 64 : COUNT = 794
CUBE = 502459875 : PATTERN = 125 : COUNT = 795
CUBE = 504358336 : PATTERN = 216 : COUNT = 796
CUBE = 506261573 : PATTERN = 343 : COUNT = 797
CUBE = 508169592 : PATTERN = 512 : COUNT = 798
CUBE = 510082399 : PATTERN = 729 : COUNT = 799
CUBE = 512000000 : PATTERN = 0 : COUNT = 800
CUBE = 513922401 : PATTERN = 1 : COUNT = 801
CUBE = 515849608 : PATTERN = 8 : COUNT = 802
CUBE = 517781627 : PATTERN = 27 : COUNT = 803
CUBE = 519718464 : PATTERN = 64 : COUNT = 804
CUBE = 521660125 : PATTERN = 125 : COUNT = 805
CUBE = 523606616 : PATTERN = 216 : COUNT = 806
CUBE = 525557943 : PATTERN = 343 : COUNT = 807
CUBE = 527514112 : PATTERN = 512 : COUNT = 808
CUBE = 529475129 : PATTERN = 729 : COUNT = 809
CUBE = 531441000 : PATTERN = 0 : COUNT = 810
CUBE = 533411731 : PATTERN = 1 : COUNT = 811
CUBE = 535387328 : PATTERN = 8 : COUNT = 812
CUBE = 537367797 : PATTERN = 27 : COUNT = 813
CUBE = 539353144 : PATTERN = 64 : COUNT = 814
CUBE = 541343375 : PATTERN = 125 : COUNT = 815
CUBE = 543338496 : PATTERN = 216 : COUNT = 816
CUBE = 545338513 : PATTERN = 343 : COUNT = 817
CUBE = 547343432 : PATTERN = 512 : COUNT = 818
CUBE = 549353259 : PATTERN = 729 : COUNT = 819
CUBE = 551368000 : PATTERN = 0 : COUNT = 820
CUBE = 553387661 : PATTERN = 1 : COUNT = 821
CUBE = 555412248 : PATTERN = 8 : COUNT = 822
CUBE = 557441767 : PATTERN = 27 : COUNT = 823
CUBE = 559476224 : PATTERN = 64 : COUNT = 824
CUBE = 561515625 : PATTERN = 125 : COUNT = 825
CUBE = 563559976 : PATTERN = 216 : COUNT = 826
CUBE = 565609283 : PATTERN = 343 : COUNT = 827
CUBE = 567663552 : PATTERN = 512 : COUNT = 828
CUBE = 569722789 : PATTERN = 729 : COUNT = 829
CUBE = 571787000 : PATTERN = 0 : COUNT = 830
CUBE = 573856191 : PATTERN = 1 : COUNT = 831
CUBE = 575930368 : PATTERN = 8 : COUNT = 832
CUBE = 578009537 : PATTERN = 27 : COUNT = 833
CUBE = 580093704 : PATTERN = 64 : COUNT = 834
CUBE = 582182875 : PATTERN = 125 : COUNT = 835
CUBE = 584277056 : PATTERN = 216 : COUNT = 836
CUBE = 586376253 : PATTERN = 343 : COUNT = 837
CUBE = 588480472 : PATTERN = 512 : COUNT = 838
CUBE = 590589719 : PATTERN = 729 : COUNT = 839
CUBE = 592704000 : PATTERN = 0 : COUNT = 840
CUBE = 594823321 : PATTERN = 1 : COUNT = 841
CUBE = 596947688 : PATTERN = 8 : COUNT = 842
CUBE = 599077107 : PATTERN = 27 : COUNT = 843
CUBE = 601211584 : PATTERN = 64 : COUNT = 844
CUBE = 603351125 : PATTERN = 125 : COUNT = 845
CUBE = 605495736 : PATTERN = 216 : COUNT = 846
CUBE = 607645423 : PATTERN = 343 : COUNT = 847
CUBE = 609800192 : PATTERN = 512 : COUNT = 848
CUBE = 611960049 : PATTERN = 729 : COUNT = 849
CUBE = 614125000 : PATTERN = 0 : COUNT = 850
CUBE = 616295051 : PATTERN = 1 : COUNT = 851
CUBE = 618470208 : PATTERN = 8 : COUNT = 852
CUBE = 620650477 : PATTERN = 27 : COUNT = 853
CUBE = 622835864 : PATTERN = 64 : COUNT = 854
CUBE = 625026375 : PATTERN = 125 : COUNT = 855
CUBE = 627222016 : PATTERN = 216 : COUNT = 856
CUBE = 629422793 : PATTERN = 343 : COUNT = 857
CUBE = 631628712 : PATTERN = 512 : COUNT = 858
CUBE = 633839779 : PATTERN = 729 : COUNT = 859
CUBE = 636056000 : PATTERN = 0 : COUNT = 860
CUBE = 638277381 : PATTERN = 1 : COUNT = 861
CUBE = 640503928 : PATTERN = 8 : COUNT = 862
CUBE = 642735647 : PATTERN = 27 : COUNT = 863
CUBE = 644972544 : PATTERN = 64 : COUNT = 864
CUBE = 647214625 : PATTERN = 125 : COUNT = 865
CUBE = 649461896 : PATTERN = 216 : COUNT = 866
CUBE = 651714363 : PATTERN = 343 : COUNT = 867
CUBE = 653972032 : PATTERN = 512 : COUNT = 868
CUBE = 656234909 : PATTERN = 729 : COUNT = 869
CUBE = 658503000 : PATTERN = 0 : COUNT = 870
CUBE = 660776311 : PATTERN = 1 : COUNT = 871
CUBE = 663054848 : PATTERN = 8 : COUNT = 872
CUBE = 665338617 : PATTERN = 27 : COUNT = 873
CUBE = 667627624 : PATTERN = 64 : COUNT = 874
CUBE = 669921875 : PATTERN = 125 : COUNT = 875
CUBE = 672221376 : PATTERN = 216 : COUNT = 876
CUBE = 674526133 : PATTERN = 343 : COUNT = 877
CUBE = 676836152 : PATTERN = 512 : COUNT = 878
CUBE = 679151439 : PATTERN = 729 : COUNT = 879
CUBE = 681472000 : PATTERN = 0 : COUNT = 880
CUBE = 683797841 : PATTERN = 1 : COUNT = 881
CUBE = 686128968 : PATTERN = 8 : COUNT = 882
CUBE = 688465387 : PATTERN = 27 : COUNT = 883
CUBE = 690807104 : PATTERN = 64 : COUNT = 884
CUBE = 693154125 : PATTERN = 125 : COUNT = 885
CUBE = 695506456 : PATTERN = 216 : COUNT = 886
CUBE = 697864103 : PATTERN = 343 : COUNT = 887
CUBE = 700227072 : PATTERN = 512 : COUNT = 888
CUBE = 702595369 : PATTERN = 729 : COUNT = 889
CUBE = 704969000 : PATTERN = 0 : COUNT = 890
CUBE = 707347971 : PATTERN = 1 : COUNT = 891
CUBE = 709732288 : PATTERN = 8 : COUNT = 892
CUBE = 712121957 : PATTERN = 27 : COUNT = 893
CUBE = 714516984 : PATTERN = 64 : COUNT = 894
CUBE = 716917375 : PATTERN = 125 : COUNT = 895
CUBE = 719323136 : PATTERN = 216 : COUNT = 896
CUBE = 721734273 : PATTERN = 343 : COUNT = 897
CUBE = 724150792 : PATTERN = 512 : COUNT = 898
CUBE = 726572699 : PATTERN = 729 : COUNT = 899
CUBE = 729000000 : PATTERN = 0 : COUNT = 900
CUBE = 731432701 : PATTERN = 1 : COUNT = 901
CUBE = 733870808 : PATTERN = 8 : COUNT = 902
CUBE = 736314327 : PATTERN = 27 : COUNT = 903
CUBE = 738763264 : PATTERN = 64 : COUNT = 904
CUBE = 741217625 : PATTERN = 125 : COUNT = 905
CUBE = 743677416 : PATTERN = 216 : COUNT = 906
CUBE = 746142643 : PATTERN = 343 : COUNT = 907
CUBE = 748613312 : PATTERN = 512 : COUNT = 908
CUBE = 751089429 : PATTERN = 729 : COUNT = 909
CUBE = 753571000 : PATTERN = 0 : COUNT = 910
CUBE = 756058031 : PATTERN = 1 : COUNT = 911
CUBE = 758550528 : PATTERN = 8 : COUNT = 912
CUBE = 761048497 : PATTERN = 27 : COUNT = 913
CUBE = 763551944 : PATTERN = 64 : COUNT = 914
CUBE = 766060875 : PATTERN = 125 : COUNT = 915
CUBE = 768575296 : PATTERN = 216 : COUNT = 916
CUBE = 771095213 : PATTERN = 343 : COUNT = 917
CUBE = 773620632 : PATTERN = 512 : COUNT = 918
CUBE = 776151559 : PATTERN = 729 : COUNT = 919
CUBE = 778688000 : PATTERN = 0 : COUNT = 920
CUBE = 781229961 : PATTERN = 1 : COUNT = 921
CUBE = 783777448 : PATTERN = 8 : COUNT = 922
CUBE = 786330467 : PATTERN = 27 : COUNT = 923
CUBE = 788889024 : PATTERN = 64 : COUNT = 924
CUBE = 791453125 : PATTERN = 125 : COUNT = 925
CUBE = 794022776 : PATTERN = 216 : COUNT = 926
CUBE = 796597983 : PATTERN = 343 : COUNT = 927
CUBE = 799178752 : PATTERN = 512 : COUNT = 928
CUBE = 801765089 : PATTERN = 729 : COUNT = 929
CUBE = 804357000 : PATTERN = 0 : COUNT = 930
CUBE = 806954491 : PATTERN = 1 : COUNT = 931
CUBE = 809557568 : PATTERN = 8 : COUNT = 932
CUBE = 812166237 : PATTERN = 27 : COUNT = 933
CUBE = 814780504 : PATTERN = 64 : COUNT = 934
CUBE = 817400375 : PATTERN = 125 : COUNT = 935
CUBE = 820025856 : PATTERN = 216 : COUNT = 936
CUBE = 822656953 : PATTERN = 343 : COUNT = 937
CUBE = 825293672 : PATTERN = 512 : COUNT = 938
CUBE = 827936019 : PATTERN = 729 : COUNT = 939
CUBE = 830584000 : PATTERN = 0 : COUNT = 940
CUBE = 833237621 : PATTERN = 1 : COUNT = 941
CUBE = 835896888 : PATTERN = 8 : COUNT = 942
CUBE = 838561807 : PATTERN = 27 : COUNT = 943
CUBE = 841232384 : PATTERN = 64 : COUNT = 944
CUBE = 843908625 : PATTERN = 125 : COUNT = 945
CUBE = 846590536 : PATTERN = 216 : COUNT = 946
CUBE = 849278123 : PATTERN = 343 : COUNT = 947
CUBE = 851971392 : PATTERN = 512 : COUNT = 948
CUBE = 854670349 : PATTERN = 729 : COUNT = 949
CUBE = 857375000 : PATTERN = 0 : COUNT = 950
CUBE = 860085351 : PATTERN = 1 : COUNT = 951
CUBE = 862801408 : PATTERN = 8 : COUNT = 952
CUBE = 865523177 : PATTERN = 27 : COUNT = 953
CUBE = 868250664 : PATTERN = 64 : COUNT = 954
CUBE = 870983875 : PATTERN = 125 : COUNT = 955
CUBE = 873722816 : PATTERN = 216 : COUNT = 956
CUBE = 876467493 : PATTERN = 343 : COUNT = 957
CUBE = 879217912 : PATTERN = 512 : COUNT = 958
CUBE = 881974079 : PATTERN = 729 : COUNT = 959
CUBE = 884736000 : PATTERN = 0 : COUNT = 960
CUBE = 887503681 : PATTERN = 1 : COUNT = 961
CUBE = 890277128 : PATTERN = 8 : COUNT = 962
CUBE = 893056347 : PATTERN = 27 : COUNT = 963
CUBE = 895841344 : PATTERN = 64 : COUNT = 964
CUBE = 898632125 : PATTERN = 125 : COUNT = 965
CUBE = 901428696 : PATTERN = 216 : COUNT = 966
CUBE = 904231063 : PATTERN = 343 : COUNT = 967
CUBE = 907039232 : PATTERN = 512 : COUNT = 968
CUBE = 909853209 : PATTERN = 729 : COUNT = 969
CUBE = 912673000 : PATTERN = 0 : COUNT = 970
CUBE = 915498611 : PATTERN = 1 : COUNT = 971
CUBE = 918330048 : PATTERN = 8 : COUNT = 972
CUBE = 921167317 : PATTERN = 27 : COUNT = 973
CUBE = 924010424 : PATTERN = 64 : COUNT = 974
CUBE = 926859375 : PATTERN = 125 : COUNT = 975
CUBE = 929714176 : PATTERN = 216 : COUNT = 976
CUBE = 932574833 : PATTERN = 343 : COUNT = 977
CUBE = 935441352 : PATTERN = 512 : COUNT = 978
CUBE = 938313739 : PATTERN = 729 : COUNT = 979
CUBE = 941192000 : PATTERN = 0 : COUNT = 980
CUBE = 944076141 : PATTERN = 1 : COUNT = 981
CUBE = 946966168 : PATTERN = 8 : COUNT = 982
CUBE = 949862087 : PATTERN = 27 : COUNT = 983
CUBE = 952763904 : PATTERN = 64 : COUNT = 984
CUBE = 955671625 : PATTERN = 125 : COUNT = 985
CUBE = 958585256 : PATTERN = 216 : COUNT = 986
CUBE = 961504803 : PATTERN = 343 : COUNT = 987
CUBE = 964430272 : PATTERN = 512 : COUNT = 988
CUBE = 967361669 : PATTERN = 729 : COUNT = 989
CUBE = 970299000 : PATTERN = 0 : COUNT = 990
CUBE = 973242271 : PATTERN = 1 : COUNT = 991
CUBE = 976191488 : PATTERN = 8 : COUNT = 992
CUBE = 979146657 : PATTERN = 27 : COUNT = 993
CUBE = 982107784 : PATTERN = 64 : COUNT = 994
CUBE = 985074875 : PATTERN = 125 : COUNT = 995
CUBE = 988047936 : PATTERN = 216 : COUNT = 996
CUBE = 991026973 : PATTERN = 343 : COUNT = 997
CUBE = 994011992 : PATTERN = 512 : COUNT = 998
CUBE = 997002999 : PATTERN = 729 : COUNT = 999
CUBE = 1000000000 : PATTERN = 0 : COUNT = 1000

Last edited by AnthonyRBrown (2024-03-23 22:23:31)

Bob · 2024-03-09 21:06:47

hi AnthonyRBrown

My time in the forum goes back far enough to remember Anthony.R.Brown who was banned for rudeness. Hopefully you are a completely different person and the name similarity is just a coincidence. Also one reformed sinner is valued in heaven so, either way, let's give you the benefit of the doubt and welcome you as a new member.

Fermat includes all integer values for n, so you have a way to go as you're just attempting to prove it for n=3. Wiles proof was for all n, and covers every possible integer for a, b and c. It has been peer reviewed and accepted by mathematicians across the world so it's a bit too hasty to dismiss it as 'far too long'.

Sorry, I'm not inclined to try your code (for a similar reason). Perhaps you could add some output (small amount only needed) . This would help me to make sense of your synopsis for how your proof works.

Bob

AnthonyRBrown · 2024-03-09 21:16:39

If you are Guilty of your past sins then you may continue!

AnthonyRBrown · 2024-03-09 21:40:56

Every possible integer for a, b and c for FLT is covered! are there any whole numbers e.g (x,y,z) cube numbers where x3 + y3 = z3 greater than (2)

FLT is only about cube numbers! The Infinite Mathematical Cube Number Demonstration covers all possible cube numbers!.

Bob · 2024-03-09 22:03:17

hi Tony,

Thanks for the reply. 'Every number' implies that your program runs through them all. I'm sure it doesn't so how do you try some values and use that to show that all values behave similarly. That's why I'm hoping for some output, please.

Bob

AnthonyRBrown · 2024-03-09 22:42:28

First of it's impossible to run through them all! for any Infinite Math problem,you can never see the last case! so it is unsolvable! for this reason you can only have a Demonstration, a 100 % Proof! is not possible only a 99.999...% proof

So if we can prove the case for any Calculable example we put forward then it's the only proof we can show...

Example : Let's look at the first 3 Groups of (10) Cube numbers...

BELOW THE FIRST TEN CUBE NUMBERS.

CUBE = (1): PATTERN = (1): COUNT = 1
CUBE = (8): PATTERN = (8): COUNT = 2
CUBE = 2(7): PATTERN = 2(7): COUNT = 3
CUBE = 6(4): PATTERN = 6(4): COUNT = 4
CUBE = 12(5): PATTERN = 12(5): COUNT = 5
CUBE = 21(6): PATTERN = 21(6): COUNT = 6
CUBE = 34(3): PATTERN = 34(3): COUNT = 7
CUBE = 51(2): PATTERN = 51(2): COUNT = 8
CUBE = 72(9): PATTERN = 72(9): COUNT = 9
CUBE = 100(0): PATTERN = (0): COUNT = 10

BELOW THE NEXT TEN.

CUBE = 133(1): PATTERN = (1): COUNT = 11
CUBE = 172(8): PATTERN = (8): COUNT = 12
CUBE = 219(7): PATTERN = 2(7): COUNT = 13
CUBE = 274(4): PATTERN = 6(4): COUNT = 14
CUBE = 337(5): PATTERN = 12(5): COUNT = 15
CUBE = 409(6): PATTERN = 21(6): COUNT = 16
CUBE = 491(3): PATTERN = 34(3): COUNT = 17
CUBE = 583(2): PATTERN = 51(2): COUNT = 18
CUBE = 685(9): PATTERN = 72(9): COUNT = 19
CUBE = 800(0): PATTERN = (0): COUNT = 20

AND THE NEXT TEN ETC.!

CUBE = 926(1): PATTERN = (1): COUNT = 21
CUBE = 1064(8): PATTERN = (8): COUNT = 22
CUBE = 1216(7): PATTERN = 2(7): COUNT = 23
CUBE = 1382(4): PATTERN = 6(4): COUNT = 24
CUBE = 1562(5): PATTERN = 12(5): COUNT = 25
CUBE = 1757(6): PATTERN = 21(6): COUNT = 26
CUBE = 1968(3): PATTERN = 34(3): COUNT = 27
CUBE = 2195(2): PATTERN = 51(2): COUNT = 28
CUBE = 2438(9): PATTERN = 72(9): COUNT = 29
CUBE = 2700(0): PATTERN = (0): COUNT = 30

The three Patterns below which are all in the first 3 Groups of (10) Cube numbers above,are in all Groups of (10) Cube numbers,(My Basic program shows this calculated Randomly for as long as you want from very low to very high numbers! but again because it's not possible to calculate them all we are left with the proof,which is Show me 1 Group of (10) Cube numbers that does not have the three Patterns,then my Demonstration is False! ?
The three Patterns in the first 3 Groups of (10) Cube numbers are explained in more detail above.

PATTERN (A) = 1,8,7,4,5,6,3,2,9,0 in brackets ()

PATTERN (B) = 7,9,7,1,1,7,9,7,1 in brackets ()

PATTERN (C) = 1,4,9,6,5,6,9,4,1,0 in brackets ()

EXAMPLE BELOW : TRY TO FIND IF THE TWO NUMBERS BELOW ADD TOGETHER MAKE A CUBE NUMBER!

2685619 + 4173281 = 6858900

Following what I have said in my Demonstration,all cube numbers are made in the same way,the infinite pattern is unique in the fact that the numbers 0 to 9 are only used once as in PATTERN A, at the end of the cubes numbers,within each group of any ten! plus more proof further on.
What it is like is a Mathematical engine that produces the cube numbers in unique groups of ten,starting from the first group of ten onwards Infinitely.
The sum of the two numbers above ( 6858900 ) is not a cube number! the reason its not a cube number is because it does not fit in any of the groups of ten,that are produced!
The nearest cube number too ( 6858900 ) is ( 6859000 ) this cube number is part of the 19 th Group given below,and is just before the start of the 20 th Group.
I will explain more how the Demonstration works below!

19 th Group

CUBE = 5929741 : PATTERN A = (1)
CUBE = 6028568 : PATTERN A = (8)
CUBE = 6128487 : PATTERN A = 2(7)
CUBE = 6229504 : PATTERN A = 6(4)
CUBE = 6331625 : PATTERN A = 12(5)
CUBE = 6434856 : PATTERN A = 21(6)
CUBE = 6539203 : PATTERN A = 34(3)
CUBE = 6644672 : PATTERN A = 51(2)
CUBE = 6751269 : PATTERN A = 72(9)
CUBE = 6859000 : PATTERN A = (0)

20 th Group

CUBE = 6967871 : PATTERN A = (1)
CUBE = 7077888 : PATTERN A = (8)
CUBE = 7189057 : PATTERN A = 2(7)
CUBE = 7301384 : PATTERN A = 6(4)
CUBE = 7414875 : PATTERN A = 12(5)
CUBE = 7529536 : PATTERN A = 21(6)
CUBE = 7645373 : PATTERN A = 34(3)
CUBE = 7762392 : PATTERN A = 51(2)
CUBE = 7880599 : PATTERN A = 72(9)
CUBE = 8000000 : PATTERN A = (0)

Bob · 2024-03-09 23:01:40

hi Tony,

I feel I'm getting closer but, maybe I'm being thick here, I don't yet get it completely.

PATTERN (A) = 1,8,7,4,5,6,3,2,9,0 in brackets ()
PATTERN (B) = 7,9,7,1,1,7,9,7,1 in brackets ()
PATTERN (C) = 1,4,9,6,5,6,9,4,1,0 in brackets ()

Pattern A makes complete sense to me. That's the units digits of the first ten cube numbers and it's easy for me to see why that repeats for the next ten and the next and so on. I could devise a proof for this without needing any number crunching.

But where did pattern B come from? I'm not seeing that in the lines above. I'm missing something here, sorry.

Pattern C looks like the units digits for square numbers (6 squared ends in 6 etc) but how did it come from the cubes.

Bob

AnthonyRBrown · 2024-03-09 23:29:16

MORE INFORMATION BELOW ON HOW MY DEMONSTRATION WORKS!

The next pattern is the group of numbers that are produced, by taking away the cube numbers starting with the largest and working backwards,as in the first group below,so 1000 - 729 = (271),729 - 512 = (217),512 - 343 = (169),343 - 216 = (127),216 -125 = (91),125 - 64 = (61),64 - 27 = (37),27 - 8 = (19),8 - 1 = (7).
What we now have is the second unique pattern (B),that shows us we have a group of related cube numbers,the end numbers are made up of 7,9,7,1,1,7,9,7,1 in brackets ()

1 st Group PATTERN B

CUBE = 1
DIFFERENCE = 7 : PATTERN B = (7)
CUBE = 8
DIFFERENCE = 19 : PATTERN B = (9)
CUBE = 27
DIFFERENCE = 37 : PATTERN B = (7)
CUBE = 64
DIFFERENCE = 61 : PATTERN B = (1)
CUBE = 125
DIFFERENCE = 91 : PATTERN B = (1)
CUBE = 216
DIFFERENCE = 127 : PATTERN B = (7)
CUBE = 343
DIFFERENCE = 169 : PATTERN B = (9)
CUBE = 512
DIFFERENCE = 217 : PATTERN B = (7)
CUBE = 729
DIFFERENCE = 271 : PATTERN B = (1)
CUBE = 1000

Now we move on to pattern (C) this pattern is made by dividing the actual COUNT position,in relation to where the cube numbers are from the start onwards! NOTE: the resulting numbers are always WHOLE numbers! the end numbers are made up of 1,4,9,6,5,6,9,4,1,0 in brackets ()

1 st Group PATTERN C

CUBE = 1 : COUNT = 1 : EQUALS 1 : PATTERN C = (1)
CUBE = 8 : COUNT = 2 : EQUALS 4 : PATTERN C = (4)
CUBE = 27 : COUNT = 3 : EQUALS 9 : PATTERN C = (9)
CUBE = 64 : COUNT = 4 : EQUALS 16 : PATTERN C = (6)
CUBE = 125 : COUNT = 5 : EQUALS 25 : PATTERN C = (5)
CUBE = 216 : COUNT = 6 : EQUALS 36 : PATTERN C = (6)
CUBE = 343 : COUNT = 7 : EQUALS 49 : PATTERN C = (9)
CUBE = 512 : COUNT = 8 : EQUALS 64 : PATTERN C = (4)
CUBE = 729 : COUNT = 9 : EQUALS 81 : PATTERN C = (1)
CUBE = 1000 : COUNT = 10 : EQUALS 100 : PATTERN C = (0)

Bob · 2024-03-10 00:12:55

Ok. I understand where patterns A, B and C come from. Because you are extracting the units digit this will always happen.

eg 3 x 3 x 3 = (2)7

83 cubed = 83 x 83 x 83 so it's units digit will be obtained by 3 x 3 x 3 = (2)7 so we get 7 again.

So pattern A occurs just because of how multiplication works. As the pattern recurs so will pattern B as you're always subtracting (m-1) from m.

If you divide any cube by its count you'll get the square numbers and they also will recur for the same reason as cubes given above.

So now I've got my three patterns.

Now what? What's the next step in the proof?

Bob

AnthonyRBrown · 2024-03-10 00:29:46

MORE INFORMATION BELOW ON HOW MY DEMONSTRATION WORKS!

As I have already said,all cube numbers are made the same way using what I would like to call a MATHEMATICAL ENGINE!
the process is from the first single cube number (1) onwards.
The most important numbers are the end numbers in brackets () the reason they are the most important is because they are the end of the process for making cube numbers! they are if you like the TRUTH numbers! its a bit like the numbers go round and round mathematically,and eventually make a whole cube number!
I have put forward three patterns A,B,C that show how these TRUTH numbers are present in any group of ten cube numbers,I am again giving the 1st and 19th Groups as examples!
The size of the cube numbers never changes how these TRUTH numbers are made! another way of looking at what im saying is that the cube numbers in the 1st Group are the same as in the 19th Group by the way they are made, and both are the same as any other group! size does not affect the TRUTH number results!
So if these two groups are the same,then trying to find if any two cube numbers make a third ,within the first group! is the same as trying with any other group,then after that test! we only need to test two groups against each other,because the maximum test is two from one group,or one from each group!
Now if we want to find out if a group of any ten numbers are cube numbers,the ten numbers must pass the three tests,they must conform to the patterns A,B,C and at the same time if we want to test if any single number is a cube number,this can be done by finding if it fits in with the nine cube numbers that are related to it,as part of its own group!.

PATTERN (A) = 1,8,7,4,5,6,3,2,9,0 in brackets ()

PATTERN (B) = 7,9,7,1,1,7,9,7,1 in brackets ()

PATTERN (C) = 1,4,9,6,5,6,9,4,1,0 in brackets ()

1 st Group

CUBE = 1: PATTERN A = (1) : COUNT = 1 : EQUALS 1 : PATTERN C = (1)
DIFFERENCE = 7 : PATTERN B = (7)
CUBE = 8: PATTERN A = (8) : COUNT = 2 : EQUALS 4 : PATTERN C = (4)
DIFFERENCE = 19 : PATTERN B = (9)
CUBE = 27: PATTERN A = (7) : COUNT = 3 : EQUALS 9 : PATTERN C = (9)
DIFFERENCE = 37 : PATTERN B = (7)
CUBE = 64: PATTERN A = (4) : COUNT = 4 : EQUALS 16 : PATTERN C = (6)
DIFFERENCE = 61 : PATTERN B = (1)
CUBE = 125: PATTERN A = (5) : COUNT = 5 : EQUALS 25 : PATTERN C = (5)
DIFFERENCE = 91 : PATTERN B = (1)
CUBE = 216: PATTERN A = (6) : COUNT = 6 : EQUALS 36 : PATTERN C = (6)
DIFFERENCE = 127 : PATTERN B = (7)
CUBE = 343: PATTERN A = (3) : COUNT = 7 : EQUALS 49 : PATTERN C = (9)
DIFFERENCE = 169 : PATTERN B = (9)
CUBE = 512: PATTERN A = (2) : COUNT = 8 : EQUALS 64 : PATTERN C = (4)
DIFFERENCE = 217 : PATTERN B = (7)
CUBE = 729: PATTERN A = (9) : COUNT = 9 : EQUALS 81 : PATTERN C = (1)
DIFFERENCE = 271 : PATTERN B = (1)
CUBE = 1000: PATTERN A = (0) : COUNT = 10 : EQUALS 100 : PATTERN C = (0)

19 th Group

CUBE = 5929741 : PATTERN A = (1) : COUNT = 181 : EQUALS 32761 : PATTERN C = (1)
DIFFERENCE = 98827 : PATTERN B = (7)
CUBE = 6028568 : PATTERN A = (8) : COUNT = 182 : EQUALS 33124 : PATTERN C = (4)
DIFFERENCE = 99919 : PATTERN B = (9)
CUBE = 6128487 : PATTERN A = (7) : COUNT = 183 : EQUALS 33489 : PATTERN C = (9)
DIFFERENCE = 101017 : PATTERN B = (7)
CUBE = 6229504 : PATTERN A = (4) : COUNT = 184 : EQUALS 33856 : PATTERN C = (6)
DIFFERENCE = 102121 : PATTERN B = (1)
CUBE = 6331625 : PATTERN A = (5) : COUNT = 185 : EQUALS 34225 : PATTERN C = (5)
DIFFERENCE = 103231 : PATTERN B = (1)
CUBE = 6434856 : PATTERN A = (6) : COUNT = 186 : EQUALS 34596 : PATTERN C = (6)
DIFFERENCE = 104347 : PATTERN B = (7)
CUBE = 6539203 : PATTERN A = (3) : COUNT = 187 : EQUALS 34969 : PATTERN C = (9)
DIFFERENCE = 105469 : PATTERN B = (9)
CUBE = 6644672 : PATTERN A = (2) : COUNT = 188 : EQUALS 35344 : PATTERN C = (4)
DIFFERENCE = 106597 : PATTERN B = (7)
CUBE = 6751269 : PATTERN A = (9) : COUNT = 189 : EQUALS 35721 : PATTERN C = (1)
DIFFERENCE = 107731 : PATTERN B = (1)
CUBE = 6859000 : PATTERN A = (0) : COUNT = 190 : EQUALS 36100 : PATTERN C = (0)

THE ABOVE EXAMPLES CLEARLY SHOW ALL CUBE NUMBERS ARE MADE THE SAME WAY! SO THE METHODS USED TO SEE IF ANY TWO NUMBERS MAKE A CUBE NUMBER, WILL ALWAYS BE THE SAME! REGARDLESS OF THE SIZE OF THE NUMBERS.

Bob · 2024-03-10 05:37:01

So what I understand is this:

You number crunch all the low value cubes and show all obey the conjecture. You have also shown that the last digit property (pattern A) applies so higher value cubes follow the same pattern and therefore obey the conjecture too.

But does that work?

Here's a simpler case.

The squares also obey the pattern A rule.

But 35, 612, 613 is a Pythagorean triple.

Bob

AnthonyRBrown · 2024-03-10 06:28:49

Can you explain what you are asking?

Are you asking if 613^2 Is a Cube number or not?

Bob · 2024-03-10 20:44:17

Clearly it's not. No, my point is a proof isn't valid if you can use it to 'prove' results that are untrue.

I have posted a puzzle recently in Exercises, called Dots and Regions. I recommend you to try it.

Bob

AnthonyRBrown · 2024-03-10 20:52:44

Maybe I'm being thick here "a proof isn't valid if you can use it to 'prove' results that are untrue" Which results are untrue ?

AnthonyRBrown · 2024-03-10 20:57:42

Below is my Cube Making program above ,making Cubes up to 100000 Which shows the three Patterns A,B,C are continuous and valid for as many Cubes as a person wants to calculate,It's a shame I can't attach the file?

PATTERN (A) = 1,8,7,4,5,6,3,2,9,0

PATTERN (B) = 7,9,7,1,1,7,9,7,1

PATTERN (C) = 1,4,9,6,5,6,9,4,1,0

Below are the last 10 Groups of 10 up to 100000...

CUBE = 997002999000000 : PATTERN = 0 : COUNT = 99900
CUBE = 997032939329701 : PATTERN = 1 : COUNT = 99901
CUBE = 997062880258808 : PATTERN = 8 : COUNT = 99902
CUBE = 997092821787327 : PATTERN = 27 : COUNT = 99903
CUBE = 997122763915264 : PATTERN = 64 : COUNT = 99904
CUBE = 997152706642625 : PATTERN = 125 : COUNT = 99905
CUBE = 997182649969416 : PATTERN = 216 : COUNT = 99906
CUBE = 997212593895643 : PATTERN = 343 : COUNT = 99907
CUBE = 997242538421312 : PATTERN = 512 : COUNT = 99908
CUBE = 997272483546429 : PATTERN = 729 : COUNT = 99909
CUBE = 997302429271000 : PATTERN = 0 : COUNT = 99910
CUBE = 997332375595031 : PATTERN = 1 : COUNT = 99911
CUBE = 997362322518528 : PATTERN = 8 : COUNT = 99912
CUBE = 997392270041497 : PATTERN = 27 : COUNT = 99913
CUBE = 997422218163944 : PATTERN = 64 : COUNT = 99914
CUBE = 997452166885875 : PATTERN = 125 : COUNT = 99915
CUBE = 997482116207296 : PATTERN = 216 : COUNT = 99916
CUBE = 997512066128213 : PATTERN = 343 : COUNT = 99917
CUBE = 997542016648632 : PATTERN = 512 : COUNT = 99918
CUBE = 997571967768559 : PATTERN = 729 : COUNT = 99919
CUBE = 997601919488000 : PATTERN = 0 : COUNT = 99920
CUBE = 997631871806961 : PATTERN = 1 : COUNT = 99921
CUBE = 997661824725448 : PATTERN = 8 : COUNT = 99922
CUBE = 997691778243467 : PATTERN = 27 : COUNT = 99923
CUBE = 997721732361024 : PATTERN = 64 : COUNT = 99924
CUBE = 997751687078125 : PATTERN = 125 : COUNT = 99925
CUBE = 997781642394776 : PATTERN = 216 : COUNT = 99926
CUBE = 997811598310983 : PATTERN = 343 : COUNT = 99927
CUBE = 997841554826752 : PATTERN = 512 : COUNT = 99928
CUBE = 997871511942089 : PATTERN = 729 : COUNT = 99929
CUBE = 997901469657000 : PATTERN = 0 : COUNT = 99930
CUBE = 997931427971491 : PATTERN = 1 : COUNT = 99931
CUBE = 997961386885568 : PATTERN = 8 : COUNT = 99932
CUBE = 997991346399237 : PATTERN = 27 : COUNT = 99933
CUBE = 998021306512504 : PATTERN = 64 : COUNT = 99934
CUBE = 998051267225375 : PATTERN = 125 : COUNT = 99935
CUBE = 998081228537856 : PATTERN = 216 : COUNT = 99936
CUBE = 998111190449953 : PATTERN = 343 : COUNT = 99937
CUBE = 998141152961672 : PATTERN = 512 : COUNT = 99938
CUBE = 998171116073019 : PATTERN = 729 : COUNT = 99939
CUBE = 998201079784000 : PATTERN = 0 : COUNT = 99940
CUBE = 998231044094621 : PATTERN = 1 : COUNT = 99941
CUBE = 998261009004888 : PATTERN = 8 : COUNT = 99942
CUBE = 998290974514807 : PATTERN = 27 : COUNT = 99943
CUBE = 998320940624384 : PATTERN = 64 : COUNT = 99944
CUBE = 998350907333625 : PATTERN = 125 : COUNT = 99945
CUBE = 998380874642536 : PATTERN = 216 : COUNT = 99946
CUBE = 998410842551123 : PATTERN = 343 : COUNT = 99947
CUBE = 998440811059392 : PATTERN = 512 : COUNT = 99948
CUBE = 998470780167349 : PATTERN = 729 : COUNT = 99949
CUBE = 998500749875000 : PATTERN = 0 : COUNT = 99950
CUBE = 998530720182351 : PATTERN = 1 : COUNT = 99951
CUBE = 998560691089408 : PATTERN = 8 : COUNT = 99952
CUBE = 998590662596177 : PATTERN = 27 : COUNT = 99953
CUBE = 998620634702664 : PATTERN = 64 : COUNT = 99954
CUBE = 998650607408875 : PATTERN = 125 : COUNT = 99955
CUBE = 998680580714816 : PATTERN = 216 : COUNT = 99956
CUBE = 998710554620493 : PATTERN = 343 : COUNT = 99957
CUBE = 998740529125912 : PATTERN = 512 : COUNT = 99958
CUBE = 998770504231079 : PATTERN = 729 : COUNT = 99959
CUBE = 998800479936000 : PATTERN = 0 : COUNT = 99960
CUBE = 998830456240681 : PATTERN = 1 : COUNT = 99961
CUBE = 998860433145128 : PATTERN = 8 : COUNT = 99962
CUBE = 998890410649347 : PATTERN = 27 : COUNT = 99963
CUBE = 998920388753344 : PATTERN = 64 : COUNT = 99964
CUBE = 998950367457125 : PATTERN = 125 : COUNT = 99965
CUBE = 998980346760696 : PATTERN = 216 : COUNT = 99966
CUBE = 999010326664063 : PATTERN = 343 : COUNT = 99967
CUBE = 999040307167232 : PATTERN = 512 : COUNT = 99968
CUBE = 999070288270209 : PATTERN = 729 : COUNT = 99969
CUBE = 999100269973000 : PATTERN = 0 : COUNT = 99970
CUBE = 999130252275611 : PATTERN = 1 : COUNT = 99971
CUBE = 999160235178048 : PATTERN = 8 : COUNT = 99972
CUBE = 999190218680317 : PATTERN = 27 : COUNT = 99973
CUBE = 999220202782424 : PATTERN = 64 : COUNT = 99974
CUBE = 999250187484375 : PATTERN = 125 : COUNT = 99975
CUBE = 999280172786176 : PATTERN = 216 : COUNT = 99976
CUBE = 999310158687833 : PATTERN = 343 : COUNT = 99977
CUBE = 999340145189352 : PATTERN = 512 : COUNT = 99978
CUBE = 999370132290739 : PATTERN = 729 : COUNT = 99979
CUBE = 999400119992000 : PATTERN = 0 : COUNT = 99980
CUBE = 999430108293141 : PATTERN = 1 : COUNT = 99981
CUBE = 999460097194168 : PATTERN = 8 : COUNT = 99982
CUBE = 999490086695087 : PATTERN = 27 : COUNT = 99983
CUBE = 999520076795904 : PATTERN = 64 : COUNT = 99984
CUBE = 999550067496625 : PATTERN = 125 : COUNT = 99985
CUBE = 999580058797256 : PATTERN = 216 : COUNT = 99986
CUBE = 999610050697803 : PATTERN = 343 : COUNT = 99987
CUBE = 999640043198272 : PATTERN = 512 : COUNT = 99988
CUBE = 999670036298669 : PATTERN = 729 : COUNT = 99989
CUBE = 999700029999000 : PATTERN = 0 : COUNT = 99990
CUBE = 999730024299271 : PATTERN = 1 : COUNT = 99991
CUBE = 999760019199488 : PATTERN = 8 : COUNT = 99992
CUBE = 999790014699657 : PATTERN = 27 : COUNT = 99993
CUBE = 999820010799784 : PATTERN = 64 : COUNT = 99994
CUBE = 999850007499875 : PATTERN = 125 : COUNT = 99995
CUBE = 999880004799936 : PATTERN = 216 : COUNT = 99996
CUBE = 999910002699973 : PATTERN = 343 : COUNT = 99997
CUBE = 999940001199992 : PATTERN = 512 : COUNT = 99998
CUBE = 999970000299999 : PATTERN = 729 : COUNT = 99999
CUBE = 1000000000000000 : PATTERN = 0 : COUNT = 100000

Last edited by AnthonyRBrown (2024-03-10 21:00:33)

Bob · 2024-03-10 21:53:45

You didn't need to post all those values. I fully accept that the three patterns carry on like that for all cubes. I have a mathematical proof that means you don't need to do the number crunching to show the continued pattern. But what has that got to do with proving FLT ?

Bob

AnthonyRBrown · 2024-03-10 22:09:12

My three Patterns prove x3 + y3 <> z3 Because if none of the 10 Cube numbers in the first group of 10,when add together do not make another Cube number,which they clearly don't,then they will not in any other group of 10,because all groups of 10 are the same!

AnthonyRBrown · 2024-03-10 23:53:52

Andrew wiles proof? is not possible! 100 % for FLT

Because it's impossible to test all 3 cases! they will always be x3(?) + y3(?) = z3(?)

As soon as someone finds numbers to test like this example that does not work x3(2685619) + y3(4173281) <> z3(6858900) then it can never be the last final test! because as soon as it is tested then I can always add another random number to either x3 or y3 because they are infinite!

Last edited by AnthonyRBrown (2024-03-11 21:28:42)

Bob · 2024-03-11 00:20:53

My three Patterns prove x3 + y3 <> z3 Because if none of the 10 Cube numbers in the first group of 10,when add together do not make another Cube number,which they clearly don't,then they will not in any other group of 10,because all groups of 10 are the same!

That's exactly what I thought you were saying.

My thought is this: Is this method of proof valid? If I can do a similar thing with square numbers and show it leads to a contradiction, then something is wrong with the proof.

Firstly do the squares follow a similar pattern:

Pattern A would be 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. You will recognise this as your cube pattern C. It would be so because you get it by doing x^3/count = x^3/x = x^2.

And the pattern repeats for every set of ten squares thereafter.

As this pattern repeats for each group of ten squares doing m^2 - (m-1)^2 will also leads to a consistent pattern B.

And pattern C will be x^2/count = x^2 / x = x so this will be the same for each group of ten squares.

Now take three numbers and square them. Does first plus second = third?

I choose 5, 2 and 3

Clearly 5^2 + 2^2 is not 3^2

So by the logic of your proof I can make up three larger numbers that end in 5, 2 and 3 and they will not form a Pythagorean triple.

I choose 35, 612 and 613

So I have a Pythagorean triple which contradicts the above.

=> method of proof is not valid

Sorry,

Bob

AnthonyRBrown · 2024-03-11 00:37:49

My Demonstration is about testing Cube numbers not! square numbers!?

The Original problem stated Xn,Yn,Zn must be > 2

Last edited by AnthonyRBrown (2024-03-11 01:17:14)

Bob · 2024-03-11 02:30:43

You have misunderstood what I'm saying.

A proof when n = 3 was divised by Euler a long time ago.

One way of showing there is a flaw in a proof is to produce a counter example. But FLT is true for n=3 so no counter example exists.

But there are an infinite number of correct solutions when n = 2 so that is why I looked at the squares.

Firstly I outlined how it is possible to make the same start you did for n = 3 ie that patterns A, B and C exist.

Then I found a triple that fails to make a solution and used your idea to make a larger triple that should, if your proof works, also fail to make a triple. But it does make a triple, so I've got the necessary counter example.

If a 'proof' is used to produce a wrong result, then the proof itself must be flawed.

https://en.wikipedia.org/wiki/Proof_of_ … _exponents

Bob

ps. I see no point in continuing this thread as we seem to have reached a dead end.

AnthonyRBrown · 2024-03-11 03:00:46

Below are the Three Patterns for the first 10 Cube numbers...

Using Patterns A,B,C below...

PATTERN (A) = 1,8,7,4,5,6,3,2,9,0

PATTERN (B) = 7,9,7,1,1,7,9,7,1

PATTERN (C) = 1,4,9,6,5,6,9,4,1,0

CUBE = 1: PATTERN A = (1) : COUNT = 1 : EQUALS 1 : DIFFERENCE = 7 : PATTERN B = (7) : PATTERN C = (1)

CUBE = 8: PATTERN A = (8) : COUNT = 2 : EQUALS 4 : DIFFERENCE = 19 : PATTERN B = (9) : PATTERN C = (4)

CUBE = 27: PATTERN A = (7) : COUNT = 3 : EQUALS 9 : DIFFERENCE = 37 : PATTERN B = (7) : PATTERN C = (9)

CUBE = 64: PATTERN A = (4) : COUNT = 4 : EQUALS 16 : DIFFERENCE = 61 : PATTERN B = (1) : PATTERN C = (6)

CUBE = 125: PATTERN A = (5) : COUNT = 5 : EQUALS 25 : DIFFERENCE = 91 : PATTERN B = (1) : PATTERN C = (5)

CUBE = 216: PATTERN A = (6) : COUNT = 6 : EQUALS 36 : DIFFERENCE = 127 : PATTERN B = (7) : PATTERN C = (6)

CUBE = 343: PATTERN A = (3) : COUNT = 7 : EQUALS 49 : DIFFERENCE = 169 : PATTERN B = (9) : PATTERN C = (9)

CUBE = 512: PATTERN A = (2) : COUNT = 8 : EQUALS 64 : DIFFERENCE = 217 : PATTERN B = (7) : PATTERN C = (4)

CUBE = 729: PATTERN A = (9) : COUNT = 9 : EQUALS 81 : DIFFERENCE = 271 : PATTERN B = (1) : PATTERN C = (1) :

CUBE = 1000: PATTERN A = (0) : COUNT = 10 : EQUALS 100 : PATTERN C = (0)

Can you please show the Group of (10) numbers your numbers below belong to that follow the Three Patterns A,B,C ?

35, 612 and 613

phrontister · 2024-03-11 21:52:14

Hi A.R.B.;

I ran your BASIC program from the code you gave in post #2...

...and here's one of the results:

Xn3 = 475647962616
Yn3 = 603351125
Zn3 = 74884638375

I had expected Zn3 to be 476251313741 (ie, the sum of Xn3 and Yn3), but it isn't. That's because your 'RANDOM NUMBER ROUTINE' finds random numbers for all three variables Xn1, Yn1 & Zn1.

This means that Zn3's value isn't related to those of the other two variables, and in many cases is less than the sum of the other two cubes! Therefore, and given the infinite range of random numbers involved, I think there's less than about 1/∞ chance that your program will ever find an occasion of Xn3 + Yn3 = Zn3.

Maybe if your program started looking for an instance of an integer cube root of (Xn3 + Yn3) it would have a greater chance of success...but probably not, coz this is the realm of infinity!

Last edited by phrontister (2024-06-01 23:40:05)

Bob · 2024-03-11 21:55:44

Bob

Math Is Fun Forum

#1 2024-03-09 05:08:37

Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#2 2024-03-09 05:10:33

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#3 2024-03-09 21:06:47

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#4 2024-03-09 21:16:39

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#5 2024-03-09 21:40:56

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#6 2024-03-09 22:03:17

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#7 2024-03-09 22:42:28

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#8 2024-03-09 23:01:40

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#9 2024-03-09 23:29:16

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#10 2024-03-10 00:12:55

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#11 2024-03-10 00:29:46

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#12 2024-03-10 05:37:01

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#13 2024-03-10 06:28:49

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#14 2024-03-10 20:44:17

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#15 2024-03-10 20:52:44

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#16 2024-03-10 20:57:42

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#17 2024-03-10 21:53:45

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#18 2024-03-10 22:09:12

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#19 2024-03-10 23:53:52

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#20 2024-03-11 00:20:53

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#21 2024-03-11 00:37:49

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#22 2024-03-11 02:30:43

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#23 2024-03-11 03:00:46

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#24 2024-03-11 21:52:14

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

#25 2024-03-11 21:55:44

Re: Fermats Last Theorem Demonstration! by, Anthony.R.Brown (1998 Solved )

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