Topologic even squares

 

By reading the sentence of Fermat explaining that his margin was too small to give the explanation of its guess  I based the hypothse that Fermat physically wanted to increase the size of his sheet.

I thus took a square of paper and proceeded to the cutting (on this left)

 

by reading its papers and his correspondence, it's seems  that he's in habit of the mistery but he always founds cleverly solutions,

 

Simple even cut  (multiple by 4)

 Cutting a square in four strips and assemble thoses one as showned at the left side.

Origin square =(n*4) named B (fb(n)) n=1 ,

(n x 4) = 16 (= 4)

C=5, A = 3

 B=C-A ou A=C-B => f(a) => f(c)-f(b)

 

for N  = 4 (formulae gived for sample )

 

Calculation of B (ou A) :

 

 

Calculation de C : (new big square)

C = fc(n)

 (4(1/4+1) , 4*(1+1/4), 5

 

 Calculation of A (ou B)

A=fa(n)

 

 

 N  divided by4

 

Calculation of B (ou A) :

 

 

Calculation of C : (new big square)

C = fc(n)

ou

 

Calculation de A (ou B)

A=fa(n)

N=6 ou impair * npair

Sample n=6:

C=9+1 = 10 or

Matrice:

N C B(or A) A(or B) C B A B+A
4 5 4

3

25 16 9 25
6 10 6 8 100 36 64 100
8 17 8 15 289 64 225 289
10 26 10 24 676 100 576 676
12 37 12 35 1369 144 1225 1369
14 50 14 48 2500 196 2304 2500
16 65 16 63 4225 256 3969 4225
18 82 18 80 6724 324 6400 6724
20 101 20 99 10201 400 9801 10201
22 122 22 120 14884 484 14400 14884
24 145 24 143 21025 576 20449 21025
26 170 26 168 28900 676 28224 28900
28 197 28 195 38809 784 38025 38809
30 226 30 224 51076 900 50176 51076
32 257 32 255 66049 1024 65025 66049
34 290 34 288 84100 1156 82944 84100

 Coefficiented Division  (k) from an even square divided by 4

 

n=1, k=2 = (4nκ) , arranged side by side

Calculation of B

Calculation of C

Calculation of A

factorised by k

 

factorised by k

 

 

 

K=1

K=2

K=3

K=4

N C B A C B A C B A C B A
4 5 3 4 10 6 8 15 9 12 20 12 16
6 10 8 6 20 16 12 30 24 18 40 32 24
8 17 15 8 34 30 16 51 45 24 68 60 32
10 26 24 10 52 48 20 78 72 30 104 96 40
12 37 35 12 74 70 24 111 105 36 148 140 48
14 50 48 14 100 96 28 150 144 42 200 192 56
16 65 63 16 130 126 32 195 189 48 260 252 64
18 82 80 18 164 160 36 246 240 54 328 320 72
20 101 99 20 202 198 40 303 297 60 404 396 80
22 122 120 22 244 240 44 366 360 66 488 480 88
24 145 143 24 290 286 48 435 429 72 580 572 96
26 170 168 26 340 336 52 510 504 78 680 672 104
28 197 195 28 394 390 56 591 585 84 788 780 112
30 226 224 30 452 448 60 678 672 90 904 896 120
32 257 255 32 514 510 64 771 765 96 1028 1020 128
34 290 288 34 580 576 68 870 864 102 1160 1152 136
36 325 323 36 650 646 72 975 969 108 1300 1292 144
38 362 360 38 724 720 76 1086 1080 114 1448 1440 152
40 401 399 40 802 798 80 1203 1197 120 1604 1596 160
42 442 440 42 884 880 84 1326 1320 126 1768 1760 168
44 485 483 44 970 966 88 1455 1449 132 1940 1932 176
46 530 528 46 1060 1056 92 1590 1584 138 2120 2112 184
48 577 575 48 1154 1150 96 1731 1725 144 2308 2300 192
50 626 624 50 1252 1248 100 1878 1872 150 2504 2496 200
52 677 675 52 1354 1350 104 2031 2025 156 2708 2700 208
54 730 728 54 1460 1456 108 2190 2184 162 2920 2912 216
56 785 783 56 1570 1566 112 2355 2349 168 3140 3132 224
58 842 840 58 1684 1680 116 2526 2520 174 3368 3360 232
60 901 899 60 1802 1798 120 2703 2697 180 3604 3596 240
62 962 960 62 1924 1920 124 2886 2880 186 3848 3840 248
64 1025 1023 64 2050 2046 128 3075 3069 192 4100 4092 256
66 1090 1088 66 2180 2176 132 3270 3264 198 4360 4352 264
68 1157 1155 68 2314 2310 136 3471 3465 204 4628 4620 272
70 1226 1224 70 2452 2448 140 3678 3672 210 4904 4896 280
72 1297 1295 72 2594 2590 144 3891 3885 216 5188 5180 288

conclusion: k parameter is a linear common factor of a,b,c

Forward:

Topologic odd squares

 

Patrick Stoltz le 19/02/2009 INPI deposit n: 343319 -

patrick.stoltz@cegetel.net

IndexTopologic even squaresFermat simpliest solutionFermat's small theoremFermat polygonals...Mathematics toolsContact me

 

-Patrick Stoltz Dpot INPI N 343319

patrick.stoltz@cegetel.net