Chapter 4 / 5 combined

Concise version - results only

e(1,3,2) = 1 : 23 = 32 - 1

e(1,5,2) £ 3 : 15 = 22 - 3, else 25 = 62 - 4

e(1,7,2) £ 7 : 27 = 112 + 7

e(2,2,2) = 0 : 32 + 42 = 52 (Pythagoras)

e(2,3,2) = 0 : 23 + 13 = 32 (smallest), 92653 + 21843 = 8976232 (largest)

e*(2,3,2) = 0 : 83 - 73 = 132 (smallest), 96323 - 27433 = 9343312 (largest)

e(2,4,2) = 1 : 74 + 54 = 552 + 1 (smallest), 99014 + 53334 = 1020721612 + 1 (largest)

e*(2,4,2) = 1 : 34 - 24 = 82 + 1 (smallest), 99614 - 28734 = 988775992 - 1 (largest)

e(2,5,2) £ 2 : 35 + 35 = 222 + 2 or 1115 + 875 = 1477662 + 2

e*(2,5,2) £ 1 : 705 - 565 = 336152 - 1

e(2,6,2) £ 1 : 316 + 296 = 385012 + 1

e*(2,6,2) £ 1 : 166 - 146 = 30412 - 1, 786 - 236 = 4743962 - 1

e(2,7,2) £ 7 : 47 + 47 = 1812 + 7

e*(2,7,2) £ 1 : 647 - 637 = 6775442 + 1

e(3,3,2) = 0 : 33 + 23 + 13 = 62 (smallest), 143 + 93 + 23 = 592, 253 + 243 + 233 = 2042

e*(3,3,2) = 0 : 33 - 13 - 13 = 52 and 53 + 33 - 23 = 122

e(3,4,2) = 0 : 204 + 154 + 124 = 4812

e*(3,4,2) = 0 : 34 - 24 - 14 = 82, 74 + 54 - 14 = 552 and 84 + 74 - 44 = 792

e(3,5,2) = 0 : 695 + 495 + 35 = 429712, 815 + 685 + 205 = 703132

e*(3,5,2) = 0 : 155 + 55 - 145 = 4742, 165 + 35 - 155 = 5382 and 585 + 215 - 335 = 249262

e(3,6,2) = 0 : 1006 + 816 + 426 = 11348652

e*(3,6,2) = 0 : 46 + 36 - 26 = 692 (smallest), 576 - 446 - 286 = 1629672

e(3,7,2) £ 1 : 47 + 27 + 27 = 1292 - 1 (parameterised solution), else 57 + 47 + 47 = 3332 + 4

e*(3,7,2) = 0 : 167 - 97 - 77 = 162122 and 647 - 637 - 17 = 6775442

e(4,7,2) = 0 : 47 + 27 + 27 + 17 = 1292, 747 + 597 + 177 + 47 = 38262702, 787 + 517 + 427 + 257 = 43243062

e(1,5,3) £ 5 : 25 = 33 + 5

e(1,7,3) £ 3 : 27 = 53 + 3

e(2,2,3) = 0 : 112 + 22 = 53 (smallest), 98302 + 11592 = 4613 (largest)

e*(2,2,3) = 0 : 32 - 12 = 23 (smallest)

e(2,3,3) = 1 : 103 + 93 = 123 + 1 and 83 + 63 = 93 - 1 (smallest)

e(2,4,3) £ 5 : 24 + 24 = 33 + 5

e*(2,4,3) £ 1 : 34 - 24 = 43 + 1

e(2,5,3) £ 7 : 55 + 35 = 153 - 7

e*(2,5,3) £ 4 : 25 - 15 = 33 + 4, else 35 - 25 = 63 - 5

e(2,6,3) £ 3 : 26 + 26 = 53 + 3

e*(2,6,3) : no improvement : 46 - 36 = 153 - 8

e(2,7,3) £ 4 : 27 + 17 = 53 + 4, else 37 + 17 = 133 - 9

e*(2,7,3) £ 2 : 27 - 17 = 53 + 2

e(3,3,3) = 0 : 53 + 43 + 33 = 63 (smallest)

e(3,4,3) = 0 : 54 + 54 + 34 = 113

e*(3,4,3) = 0 : 34 - 24 - 14 = 43, 94 - 84 - 74 = 43 and 314 + 204 - 174 = 1003 (largest)

e(3,5,3) £ 3 : 815 + 745 + 155 = 17873 - 3

e*(3,5,3) £ 1 : 925 + 335 - 745 = 16403 + 1 and 695 - 675 - 95 = 5983 + 1

e(3,6,3) £ 2 : 256 + 236 + 66 = 7323 + 2

e*(3,6,3) = 0 : 196 - 186 - 156 = 1183

e(3,7,3) £ 8 : 37 + 17 + 17 = 133 - 8, else 157 + 57 + 47 = 5553 + 9

e*(3,7,3) £ 1 : 27 - 17 - 17 = 53 + 1

e(4,5,3) = 0 : 45 + 35 + 25 + 25 = 113 (smallest), 485 + 85 + 55 + 35 = 6343 (largest)

e*(4,5,3) = 0 : 225 + 195 + 135 - 45 = 2003 and 255 + 255 + 135 - 25 = 2713

e(4,6,3) £ 3 : 576 + 196 + 196 + 106 = 32523 + 3 and 686 + 536 + 256 + 156 = 49503 + 3

e*(4,6,3) £ 1 : 86 - 56 - 46 - 46 = 623 - 1 and 66 + 26 - 56 - 46 = 303 - 1

e(4,7,3) £ 7 : 37 + 17 + 17 + 17 = 133 - 7, else 107 + 67 + 57 + 37 = 2183 + 16

e*(4,7,3) = 0 : 27 - 17 - 17 - 17 = 53, else 297 + 237 + 117 - 157 = 27373 - 1

e(5,6,3) £ 2 : 286 + 216 + 156 + 106 + 66 = 8343 + 2 and 286 + 216 + 166 + 166 + 36 = 8443 + 2

e(5,7,3) £ 6 : 37 + 17 + 17 + 17 + 17 = 133 - 6

e(6,6,3) £ 3 : 106 + 96 + 76 + 76 + 46 + 36 = 1213 + 3 (smallest)

e(6,7,3) £ 1 : 207 + 77 + 27 + 27 + 27 + 27 = 10863 - 1

e(7,6,3) = 0 : 306 + 246 + 96 + 96 + 36 + 36 + 16 = 9733

e(7,7,3) = 0 : 207 + 77 + 27 + 27 + 27 + 27 + 17 = 10863

e(1,3,4) £ 11 : 33 = 24 + 11

e(1,5,4) £ 13 : 35 = 44 - 13

e(1,7,4) £ 47 : 27 = 34 + 47

e(2,2,4) = 0 : 242 + 72 = 54, 1202 + 1192 = 134 and 2402 + 1612 = 174 (from Pythagoras)

e*(2,2,4) = 0 : 52 - 32 = 24 (smallest)

e(2,3,4) £ 3 : 633 + 313 = 234 - 3

e*(2,3,4) £ 1 : 213 - 193 = 74 + 1 and 373 - 33 = 154 + 1

e(2,4,4) £ 46 : 54 + 54 = 64 - 46

e(2,5,4) £ 12 : 35 + 15 = 44 - 12, else 25 + 25 = 34 - 17

e*(2,5,4) : no improvement

e(2,6,4) £ 16 : 26 + 16 = 34 - 16

e*(2,6,4) : no improvement

e(2,7,4) £ 86 : 37 + 27 = 74 - 86

e*(2,7,4) £ 33 : 67 - 27 = 234 - 33

e(3,3,4) = 0 : 203 + 173 + 123 = 114 and 743 + 483 + 253 = 274

e*(3,3,4) = 0 : 153 + 93 - 23 = 84 and 113 - 33 - 23 = 64

e(3,4,4) = 0 : 4145604 + 2175194 + 958004 = 4224814 (Frye), else 474 + 224 + 214 = 484 + 2

e(3,5,4) £ 3 : 45 + 35 + 25 = 64 + 3

e*(3,5,4) : no improvement : 165 + 135 - 175 = 24 - 4

e(3,6,4) £ 15 : 26 + 16 + 16 = 34 - 15

e*(3,6,4) : no improvement

e(3,7,4) £ 22 : 47 + 37 + 37 = 124 + 22

e*(3,7,4) : no improvement

e(4,5,4) £ 3 : 1045 + 725 + 525 + 455 = 3484 - 3

e*(4,5,4) £ 1 : 95 + 55 - 75 - 75 = 134 - 1, 165 + 135 + 35 - 175 = 44 - 1

e(4,6,4) £ 10 : 216 + 186 + 116 + 36 = 1054 + 10

e*(4,6,4) £ 1 : 66 + 46 - 26 - 26 = 154 - 1

e(4,7,4) £ 16 : 97 + 57 + 47 + 37 = 474 - 16

e*(4,7,4) £ 5 : 297 + 107 + 67 - 257 = 3254 - 5

e(5,5,4) = 0 : 1855 + 1695 + 1095 + 465 + 15 = 7804

e(5,6,4) £ 3 : 96 + 86 + 56 + 36 + 26 = 304 + 3

e(5,7,4) £ 7 : 317 + 147 + 137 + 117 + 107 = 4084 + 7

e(6,6,4) £ 1 : 96 + 86 + 76 + 46 + 46 + 46 = 314 + 1

e(6,7,4) £ 8 : 317 + 147 + 137 + 117 + 107 + 17 = 4084 + 8, else 167 + 137 + 107 + 77 + 57 + 47 = 1364 + 9

e(7,6,4) = 0 : 926 + 666 + 576 + 506 + 486 + 226 + 86 = 9314

e(1,7,5) £ 115 : 27 = 35 - 115

e(2,2,5) = 0 : 412 + 382 = 55 (smallest), 61212 + 56462 = 375 (largest)

e*(2,2,5) = 0 : 92 - 72 = 25

e(2,3,5) £ 2 : 2713 + 2393 = 325 - 2

e*(2,3,5) £ 2 : 373 - 353 = 65 + 2

e(2,4,5) £ 65 : 224 + 114 = 125 + 65

e*(2,4,5) £ 1 : 374 - 344 = 145 + 1

e(2,5,5) £ 12 : 165 + 135 = 175 + 12

e(2,6,5) £ 115 : 26 + 26 = 35 - 115

e*(2,6,5) : no improvement

e(2,7,5) £ 13 : 27 + 27 = 35 + 13

e*(2,7,5) : no improvement

e(3,3,5) = 0 : 773 + 573 + 493 = 155 and 863 + 403 + 393 = 155

e*(3,3,5) = 0 : 173 + 153 - 83 = 65, 413 - 333 - 63 = 85 and 483 - 213 - 113 = 105

e(3,4,5) £ 1 : 804 + 664 + 274 = 365 + 1

e*(3,4,5) = 0 : 74 - 64 - 34 = 45 and 374 - 344 - 14 = 145

e(3,5,5) £ 18 : 135 + 135 + 75 = 155 + 18

e(3,6,5) £ 51 : 26 + 26 + 26 = 35 - 51

e*(3,6,5) £ 9 : 316 - 216 - 176 = 605 - 9

e(3,7,5) £ 14 : 27 + 27 + 17 = 35 + 14, else 27 + 17 + 17 = 35 - 113

e*(3,7,5) £ 1 : 197 + 77 - 187 = 495 + 1

e(4,4,5) = 0 : 334 + 334 + 174 + 124 = 195 and 374 + 264 + 194 + 114 = 195

e(4,5,5) = 0 : 1335 + 1105 + 845 + 275 = 1445 (Lander & Parkin)

e(4,6,5) £ 13 : 26 + 26 + 26 + 26 = 35 + 13

e*(4,6,5) £ 8 : 316 + 16 - 216 - 176 = 605 - 8

e(4,7,5) £ 15 : 27 + 27 + 17 + 17 = 35 + 15, else 47 + 27 + 27 + 27 = 75 - 39

e*(4,7,5) = 0 : 197 + 77 + 187 - 17 = 495 (from k = 3)

e(5,6,5) £ 4 : 56 + 56 + 36 + 36 + 26 = 85 + 4

e(5,7,5) £ 16 : 27 + 27 + 17 + 17 + 17 = 35 + 16, else 47 + 27 + 27 + 27 + 17 = 75 - 38

e(6,6,5) £ 5 : 56 + 56 + 36 + 36 + 26 + 16 = 85 + 5, else 86 + 66 + 56 + 56 + 56 + 56 = 135 + 7

e(6,7,5) £ 17 : 27 + 27 + 17 + 17 + 17 + 17 = 35 + 17, else 47 + 27 + 27 + 27 + 17 + 17 = 75 - 37

e(7,6,5) = 0 : 906 + 856 + 686 + 686 + 576 + 546 + 196 = 2595

e(7,7,5) = 0 : 467 + 457 + 327 + 237 + 127 + 77 + 67 = 2435

e(1,7,6) £ 759 : 47 = 56 + 759

e(2,2,6) = 0 : 1172 + 442 = 56 , 20352 + 8282 = 136 and 48882 + 4952 = 176 (from n = 3)

e*(2,2,6) = 0 : 172 - 152 = 26

e(2,3,6) = 1 : 83 + 63 = 36 - 1

e*(2,3,6) = 1 : 123 - 103 = 36 - 1

e(2,4,6) £ 23 : 54 + 34 = 36 - 23

e*(2,4,6) £ 1 : 34 - 24 = 26 + 1

e(2,5,6) £ 53 : 55 + 45 = 46 + 53

e*(2,5,6) £ 52 : 45 - 35 = 36 + 52

e(2,6,6) £ 601 : 26 + 26 = 36 - 601, else 86 + 86 = 96 - 7153

e(2,7,6) £ 278 : 37 + 37 = 46 + 278

e*(2,7,6) : no improvement

e(3,3,6) = 0 : 83 + 63 + 13 = 36 and 223 + 173 + 43 = 56 (from n = 3)

e*(3,3,6) = 0 : 123 + 13 - 103 = 36, 153 + 93 - 23 = 46 and 63 - 53 - 33 = 26

e(3,4,6) £ 7 : 54 + 34 + 24 = 36 - 7

e*(3,4,6) = 0 : 314 + 204 - 174 = 106 and 894 - 734 - 244 = 186

e(3,5,6) £ 41 : 65 + 65 + 25 = 56 - 41

e*(3,5,6) £ 20 : 45 - 35 - 25 = 36 + 20

e(3,6,6) £ 219 : 56 + 56 + 56 = 66 + 219

e(3,7,6) £ 279 : 37 + 37 + 17 = 46 + 279, else 27 + 27 + 27 = 36 - 345

e*(3,7,6) £ 150 : 37 + 37 - 27 = 46 + 150

e(4,4,6) = 0 : 614 + 564 + 264 + 44 = 624 + 474 + 464 + 84 = 674 + 444 + 224 + 84 = 176

e(4,5,6) £ 2 : 195 + 125 + 115 + 105 = 126 - 2

e*(4,5,6) : no improvement

e(4,6,6) £ 430 : 126 + 116 + 116 + 106 = 146 - 430

e(4,7,6) £ 217 : 27 + 27 + 27 + 27 = 36 - 217

e*(4,7,6) £ 22 : 37 + 37 - 27 - 27 = 46 + 22

e(5,4,6) = 0 : 164 + 144 + 104 + 74 + 64 = 184 + 104 + 64 + 64 + 34 = 184 + 104 + 74 + 44 + 24 = 76.

e(5,5,6) £ 1 : 55 + 35 + 35 + 35 + 35 = 46 + 1 and 195 + 125 + 115 + 105 + 15 = 126 - 1

e(5,6,6) £ 299 : 126 + 116 + 116 + 106 + 36 = 146 + 299

e(5,7,6) £ 89 : 27 + 27 + 27 + 27 + 27 = 36 - 89

e(6,5,6) = 0 : 195 + 125 + 115 + 105 + 15 + 15 = 126 and 1195 + 815 + 465 + 415 + 245 + 145 = 556

e(6,6,6) £ 3 : 106 + 96 + 76 + 76 + 46 + 36 = 116 + 3

e(6,7,6) £ 39 : 27 + 27 + 27 + 27 + 27 + 27 = 36 - 39

e(7,6,6) = 0 : (10776 + 8946 + 7026 + 4746 + 4026 + 2346 + 746 = 11416 (Lander & Parkin)

e(2,2,7) = 0 : 2782 + 292 = 57

e*(2,2,7) = 0 : 332 - 312 = 27

e(2,3,7) £ 2 : 53 + 13 = 27 - 2

e*(2,3,7) £ 1 : 73 - 63 = 27 - 1

e(2,4,7) £ 14 : 234 + 34 = 167 - 14

e*(2,4,7) : no improvement

e(2,5,7) £ 139 : 45 + 45 = 37 - 139

e*(2,5,7) £ 83 : 35 - 25 = 27 + 83

e(2,6,7) £ 30 : 56 + 36 = 47 - 30

e(2,7,7) £ 1931 : 27 + 27 = 37 - 1931, else 57 + 57 = 67 - 123686

e(3,3,7) £ 1 : 53 + 13 + 13 = 27 - 1, 113 + 83 + 73 = 37 - 1 and 1563 + 803 + 783 = 97 - 1

e*(3,3,7) = 0 : 73 - 63 + 13 = 27, 753 - 523 - 113 = 67 and 853 - 773 - 543 = 27

e(3,4,7) £ 2 : 234 + 34 + 24 = 67 + 2

e*(3,4,7) : no improvement

e(3,5,7) £ 7 : 215 + 145 + 115 = 97 + 7

e*(3,5,7) : no improvement

e(3,6,7) £ 29 : 56 + 36 + 16 = 47 - 29

e*(3,6,7) : no improvement

e(3,7,7) £ 1803 : 27 + 27 + 27 = 37 - 1803, else 37 + 37 + 37 = 47 - 9823

e(4,3,7) = 0 : 53 + 13 + 13 + 13 = 27, 113 + 83 + 73 + 13 = 37 and 563 + 463 + 193 + 53 = 67

e(4,4,7) = 0 : 1264 + 1114 + 504 + 224 = 177

e*(4,4,7) = 0 : 174 - 144 - 144 - 94 = 27

e(4,5,7) £ 8 : 215 + 145 + 115 + 15 = 97 + 8, else 45 + 45 + 25 + 25 = 37 - 75

e*(4,5,7) £ 1 : 285 + 265 + 115 - 255 = 117 - 1

e(4,6,7) £ 28 : 56 + 36 + 16 + 16 = 47 - 28

e*(4,6,7) : no improvement

e(4,7,7) £ 1675 : 27 + 27 + 27 + 27= 37 - 1675, else 37 + 37 + 37 + 37 = 47 - 7636

e(5,4,7) £ 1 : 604 + 454 + 354 + 314 + 74 = 117 + 1

e(5,5,7) = 0 : 5055 + 3155 + 2665 + 2125 + 1155 = 877

e(5,6,7) £ 13 : is 446 + 386 + 246 + 106 + 106 = 277 + 13

e(5,6,7) £ 1547 : 27 + 27 + 27 + 27 + 27 = 37 - 1547, else 97 + 97 + 67 + 57 + 57 = 107 + 2124

e(6,6,7) £ 4 : 246 + 236 + 196 + 176 + 56 + 36 = 177 - 4

e(6,7,7) £ 1419 : 27 + 27 + 27 + 27 + 27 + 27 = 37 - 1419

The best estimates for S(m,n) based on the above data are presented in the following table. An asterisk indicates that equality has not been proved.

 n\m 2 3 4 5 6 7 2 2 2 3 3 * 3 * 4 * 3 2 3 3 * 4 * 7 * 7 * 4 2 3 * 3 5 * 7 * 8 * 5 2 3 * 4 * 4 * 7 * 7 * 6 2 3 4 * 6 * 7 * 9 * 7 2 4 * 4 * 5 * 8 * 8 *

Finally, the following table summarises S*(m,n).

 n\m 2 3 4 5 6 7 2 2 2 3 3 3 3 3 2 3 3 4 3 4 4 2 3 3 5 5 8 5 2 3 3 4 7 4 6 2 3 3 6 7 9 7 2 3 4 5 8 8