Exponential diophantine problems Gottfried Helms - Univ Kassel          (Miniatures since 07' 2007)

Table for Fermat-quotients of degree m≥2
for integer bases b<p (where p
∈ ℙ )

The following table displays p-adic-expansions of bases b, for which

bp-1 ≡ 1 (mod pm)

In ([1],2009) I discussed earlier the problem of Fermat-quotients and introduced a couple of small tools and concepts which I use here. This table focuses on data for only a specific subset of the entire problem.

The casual reader who is not aware of the concept of Fermat-quotient may look in the wikipedia (see:[wp]) first which gives some basic insight and a couple of further references/online-links.

The reader may find some quite involved related tables in the internet, see the references-section at the end of this article for some online-links to homepages which I got aware of.

Some preliminary definitions:

1) In the following I (mis)use the braces for a shorter notation to express the above statement for a more general view. (A misuse which has been helpful through all my discussions/treatizes of exponential diophantine problems.) For the case, that a number n contains the primefactor p to the m'th power I write for p's canonical exponent in n:

{n, p} = m                                          ==> n = x ∙ pm ∙with gcd(x,p)=1

Note that this notation is a bit more algebraic/concise than "pm| n" or "n≡0 (mod pm)"because -if in the two latter two cases m is the indeterminate- then the expression gives only an upper limit and not a precise value to m. So for instance,

{2p-1-1, p} = 2

is a bit more restricted/precise than the original definition for a Wieferich-prime p which requires only

{2p-1-1, p} 2

(where we know only two solutions p=1093 and p=3511 for m ≥2 anyway).

Similary for instance

{35-1 , 11} = 2
{19
6-1 , 7} = 3
{740862
6 – 1,7}  = 7
{175909088838
12 – 1,13} = 12

indicate Fermat-quotients for (3, 11), (19, 7), (740862, 7) and (175909088838, 13) of degrees 2,3,7 and 12 respectively.

2) Let's define here the term "degree of the Fermat-quotient":

{bp-1-1, p } = m                  ==> m is the degree of the Fermat-quotient of the pair (b,p)

3) By the p-adic-representation of the bases b which provide high Fermat-quotients for some prime p we can read the table like in the following example:

 p b d0 d1 d2 d3 d4 d4 d5 d6 d7 d8 d9 d10 ... 7 1 1 0 0 0 0 0 0 0 0 0 0 0 ... 7 2 2 4 6 3 0 2 6 2 4 3 4 4 ... 7 3 3 4 6 3 0 2 6 2 4 3 4 4 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

Here the column p indicates the prime, the column b the "root" bases - "root" means here, that from b = d0 further bases bm can be derived according to the given p-adic digits d0,d1,d2,... Actually the entry in that column is identical with the entry in column d0 which simply contain all congruence-classes (mod p) .

The "p-adic" convention means, that the "digits" are just the digits of the number-representation of some number in the number-system to base p, so to say:

bm = "dm-1 ... d2 d1 d0" base p

so for the row with "root" base b=2 we can get the following subsequent bases b1,b2,...

b1 = 2                                   = 2                         ==>        {26 -1, 7} = 1
b2 = 2 + 4∙7                        = 30                      ==>        {30
6-1, 7} = 2
b3 = 2 + 4∙7+ 6∙7
2            = 324                    ==>        {3246-1, 7} = 3
...
and in general
bm = 2 + 4∙7+ ... +dm-1∙7
m-1                            ==>        {bm6-1, 7} = m

But note, that also infinitely many bases of the form

let x a positive integer such that gcd(x,6) = 1
b2 = 2 + 4∙7+ x∙7
2            = 30+x∙49           ==>        {b26-1, 7} = 2

give also fermat-quotients of degree 2, so

{796 - 1, 7} = 2

We can conclude one more result: because we see here a set of infinitely many bases in arithmetic progression, we have even - due to Dirichlet's theorem - a proof, that for each prime p and each degree m we'll find infinitely bases bm which are prime themselves.

So such a table essentially gives the (p-1)th roots of 1 modulo some pm in the p-adic sense and can in analogy be thought as representing the unit-roots exp(î∙π∙m/(p-1)) in the case of the complex number plane.

(end of preliminary definitions)

Note that b=d0 being a residue (mod p) means, that all bases b gotten by p-adic-expansions beginning at a residue d0 <>0 (mod p) are also coprime to p and have no (yet recognizable) simple restrictions, so for instance they

*  can be even,
*  can be odd,
and in particular
*  can be prime themselves.

Sometimes tabulations of Fermat-quotients display only prime bases b - for instance because there is great relevance connected to other exponential diophantine problems like FLT -for instance by the analyses of A. Wieferich (after whom that Wieferich-primes are named) and others. But this is not intended here; here I want to display the general scheme and in that scheme I want to focus to that subset of higher Fermat-quotients where the bases are in the residue classes of the associated primes and thus smaller that that primes.

Base b smaller than prime p

The example computation in the above lines shows immediately, that the displayed bases b = b1, simply equalling the residue-classes (mod p), are the only ones which are smaller than the prime which defines the congruence. High fermat-quotients are very rare for such cases; two prominent examples are the above mentioned two Wieferich primes p=1093 and p=3511 to base 2 where we have 210921 (mod 10932) and 235101 (mod 35112) or in the new notation {21092-1, 1093} = 2 and {23510-1, 3511} = 2 ; another one is p=11 and the base b=3 where we have {310 - 1, 11} = 2 .

To have such a case is equivalent to have the digit d1=0 in the table, because that means, according to the example above

b2 = d0 + 0∙p1                     = d0        = b1        ==>        {b1p-1 -1 , p} = 2

and the earliest example is that for prime p = 11 and base b1 = 3

 p b d0 d1 d2 d3 d4 d4 d5 d6 d7 d8 d9 d10 d11 11 3 3 0 1 2 3 6 10 8 7 0 6 3 9

(Note that in general zeros in the digits occur randomly scattered over the digits in the table)

The well known Wieferich-primes p=1093 and p=3511 are immediately recognizable in the table by this: they are the only occurences so far, where the root base (= residue d0) equals 2 (or some small powers of 2) and the digit d1=0. They are marked orange here:

 p b d0 d1 d2 d3 d4 d4 d5 d6 d7 d8 d9 d10 d11 1093 2 2 0 974 227 564 478 1009 395 581 1027 834 136 534 3511 2 2 0 102 2379 495 3468 1939 2849 2386 752 3077 803 3499

Higher degrees of Fermatquotiens

For even higher Fermat-quotients we can proceed by looking for further consecutive zeros in the digits:

b3 = d0 + 0∙p1+ 0∙p2         = b0                       ==>        {b3p-1 - 1, p} = 3
bm = d0 + 0∙p
1+ ...+ 0∙pm-1= b0                      ==>        {bmp-1 - 1, p} = m
...

Cases m=3

The only case of the fermat-quotient of order m=3 (where also the base is smaller than the defining prime) occurs with the prime p=113 and the base b3 = b1 = d0 = 68 :

 p b d0 d1 d2 d3 d4 d4 d5 d6 d7 d8 d9 d10 d11 113 68 68 0 0 92 4 67 65 0 13 7 106 20 16

For this I checked the primes up to p=17 389 (which is the 2 000th prime). In the mathematical-discussionboard MSE (see[2]) one correspondent confirmed that observation and proceeded saying he had found this to be true up to p=104 729, (which is the 10 000'th prime) . In [3] we find that no further example occurs up to p ~ 3.6 ∙ 108

Cases m>3

I've not found any higher degree Fermat-quotients for pairs of bases and primes with b<p.

Gottfried Helms, 7.12.2013

Table 1, b<p:

p-adic representations for bases allowing fermat-quotients up to degree 12, where the high fermat-quotient occurs already for the "root" bases (which are smaller than the defining prime). That is equivalent to say that the p-adic-digit d1 = 0 .

 p b d0 d1 d2 d3 d4 d4 d5 d6 d7 d8 d9 d10 d11 11 3 3 0 1 2 3 6 10 8 7 0 6 3 9 11 9 9 0 6 1 9 8 7 1 3 2 0 0 6 29 14 14 0 6 9 19 22 8 24 4 3 19 16 25 37 18 18 0 26 16 3 34 26 35 26 24 18 24 1 43 19 19 0 34 35 10 37 7 26 6 41 34 4 21 59 53 53 0 28 27 40 50 43 2 38 17 21 45 38 71 11 11 0 9 23 19 43 34 58 41 12 49 52 48 71 26 26 0 19 48 7 70 7 9 25 60 45 27 29 79 31 31 0 22 69 9 4 0 69 27 67 11 51 42 97 53 53 0 87 20 43 33 35 56 75 38 17 7 71 103 43 43 0 69 94 23 81 59 78 91 45 75 19 9 109 96 96 0 63 53 40 18 51 44 40 99 57 62 83 113 68 68 0 0 92 4 67 65 0 13 7 106 20 16 127 38 38 0 14 80 89 79 92 87 33 49 34 108 78 127 62 62 0 38 18 74 120 88 26 85 117 22 102 121 131 58 58 0 58 45 43 35 118 39 49 117 20 67 93 131 111 111 0 100 6 68 71 120 87 121 113 124 2 1 137 19 19 0 32 71 56 130 42 88 50 33 95 120 36 151 78 78 0 24 93 15 23 57 16 50 119 20 54 29 163 65 65 0 117 74 136 122 122 90 95 161 79 119 141 163 84 84 0 30 38 143 65 86 83 84 150 160 147 3 181 78 78 0 122 38 129 144 116 13 64 54 10 97 32 191 176 176 0 70 66 133 24 78 95 120 16 148 13 67 197 143 143 0 180 145 6 84 55 185 7 77 174 25 41 199 174 174 0 105 162 83 129 100 104 106 36 10 83 176 211 165 165 0 192 78 98 178 173 190 77 96 101 58 86 211 182 182 0 155 11 37 210 111 113 126 10 199 119 24 223 69 69 0 41 45 55 211 154 104 52 8 140 215 115 229 44 44 0 120 127 35 174 225 143 172 7 161 17 73 229 209 209 0 219 104 7 38 62 94 32 167 209 89 92 233 33 33 0 34 13 124 218 79 170 138 23 22 145 55 241 94 94 0 200 219 44 40 139 133 194 33 79 216 98 257 48 48 0 35 193 232 136 111 42 246 8 240 108 153 263 79 79 0 228 138 101 146 179 235 11 154 235 65 25 269 171 171 0 113 237 101 88 201 166 6 182 69 230 91 269 180 180 0 74 218 219 202 244 259 140 164 82 64 245 269 207 207 0 77 22 243 185 188 90 167 129 189 99 31 281 20 20 0 127 19 267 238 270 101 2 147 168 255 241 283 147 147 0 26 218 169 270 98 171 149 208 28 73 159 293 91 91 0 39 206 171 105 26 97 289 219 38 212 97 307 40 40 0 65 32 263 223 198 59 173 201 300 51 156 313 104 104 0 218 120 107 74 51 239 108 42 125 305 227 313 213 213 0 58 53 223 63 66 120 238 69 293 236 74 331 18 18 0 14 185 36 326 146 77 236 117 314 217 208 331 71 71 0 147 280 26 271 249 28 266 190 81 72 272 331 324 324 0 173 41 188 39 158 186 190 192 261 44 66 347 75 75 0 27 292 130 202 322 311 308 91 124 68 45 347 156 156 0 204 298 48 283 227 292 96 238 158 288 171 349 223 223 0 164 173 80 185 86 312 39 96 135 319 200 349 317 317 0 103 38 68 134 72 47 119 213 260 143 75 353 14 14 0 337 259 116 221 334 144 288 263 294 249 149 353 196 196 0 258 218 347 348 162 301 246 26 14 219 298 359 257 257 0 7 62 98 138 61 136 346 355 202 95 59 359 331 331 0 220 114 233 245 242 343 204 171 157 105 184 367 159 159 0 8 121 119 104 131 157 155 171 361 260 334 367 205 205 0 357 199 82 254 320 156 227 196 115 135 257 373 242 242 0 85 355 365 313 261 262 275 8 304 76 16 379 174 174 0 198 72 194 145 375 164 136 90 120 97 280 397 175 175 0 184 332 45 282 293 362 7 18 355 77 396 401 280 280 0 111 66 332 351 28 254 130 160 12 373 20 419 369 369 0 210 182 164 36 27 57 284 65 109 78 307 421 251 251 0 391 78 129 155 80 102 361 409 247 41 368 433 349 349 0 226 95 120 215 8 94 113 137 206 50 182 439 194 194 0 171 199 141 134 421 15 420 120 321 21 35 449 210 210 0 241 129 137 282 393 314 294 83 246 101 212 461 52 52 0 206 196 154 265 320 98 291 416 232 273 451 463 255 255 0 177 75 422 113 430 83 351 42 169 375 265 463 345 345 0 40 331 112 286 24 426 256 175 307 436 370 487 10 10 0 329 462 103 341 149 54 64 287 349 100 311 487 100 100 0 249 0 258 336 428 281 36 237 54 415 322 487 175 175 0 427 44 471 72 106 273 475 187 406 475 207 487 307 307 0 424 391 356 98 429 194 148 479 236 420 3 499 346 346 0 460 279 269 135 282 230 492 301 219 312 420 509 93 93 0 263 29 65 137 86 45 367 277 294 481 493 509 250 250 0 362 34 224 121 181 23 163 373 211 115 476 521 308 308 0 234 382 126 54 143 270 374 143 129 105 504 523 241 241 0 177 363 517 57 483 214 449 291 455 515 365 547 427 427 0 80 35 1 9 519 12 248 135 52 437 525 557 430 430 0 216 407 507 519 454 443 482 394 423 307 4 563 486 486 0 95 227 453 37 221 314 138 122 430 552 234 571 262 262 0 465 32 170 218 521 229 319 188 359 219 231 571 422 422 0 235 88 536 559 257 229 106 540 414 430 452 571 516 516 0 432 328 306 112 386 407 316 168 298 190 9 577 427 427 0 173 365 178 420 563 350 442 230 281 568 135 599 559 559 0 22 516 267 499 67 485 573 452 454 565 338 599 588 588 0 589 354 135 381 250 166 129 215 282 372 246 601 405 405 0 206 198 25 241 258 187 133 391 525 404 295 607 162 162 0 28 95 427 332 475 143 10 374 525 53 61 607 302 302 0 298 328 238 332 253 427 365 482 386 364 1 617 556 556 0 295 98 413 185 346 272 288 5 439 33 555 619 286 286 0 412 73 19 138 443 323 552 101 519 576 4 631 69 69 0 436 306 380 15 431 309 438 213 508 462 250 631 534 534 0 166 492 272 428 130 153 213 346 378 261 273 641 340 340 0 262 69 543 35 618 472 390 519 613 20 474 647 56 56 0 140 27 415 35 514 425 378 583 89 458 253 647 526 526 0 424 398 143 435 400 187 561 421 578 392 523 653 84 84 0 256 398 21 227 576 63 105 604 360 551 337 653 120 120 0 380 432 266 387 96 541 640 138 311 454 151 653 197 197 0 488 104 542 530 73 535 89 26 154 125 304 653 287 287 0 242 101 139 90 595 75 26 265 559 647 617 653 410 410 0 550 517 360 607 363 295 451 73 367 504 652 659 503 503 0 332 163 234 104 573 123 3 637 498 72 46 661 184 184 0 24 88 510 39 522 472 487 446 32 600 5 673 22 22 0 131 423 461 316 512 460 298 0 549 192 316 673 484 484 0 380 449 457 280 57 208 610 668 555 341 244 701 375 375 0 608 72 366 153 324 202 483 404 383 105 355 727 92 92 0 148 465 314 508 11 31 88 13 164 581 357 739 168 168 0 338 187 646 243 602 64 62 195 681 478 230 743 467 467 0 301 187 637 366 614 176 85 187 552 288 193 761 675 675 0 240 616 172 161 269 43 462 481 759 669 550 769 392 392 0 749 733 564 221 316 264 293 466 340 696 542 773 392 392 0 575 155 542 561 413 446 173 546 81 245 375 797 440 440 0 651 772 713 596 647 154 69 511 416 563 228 797 446 446 0 133 654 744 149 541 441 195 222 529 417 1 809 207 207 0 136 315 270 374 671 514 533 322 556 253 603 823 393 393 0 546 57 76 596 160 414 133 740 627 385 297 827 314 314 0 587 431 569 179 608 385 0 118 506 728 509 827 381 381 0 471 730 400 773 607 299 760 280 656 178 480 829 46 46 0 416 576 9 262 755 19 576 659 306 652 387 829 632 632 0 805 672 312 78 524 275 738 208 683 20 821 839 667 667 0 11 295 797 257 832 360 25 153 134 401 788 857 682 682 0 580 841 472 470 270 553 763 247 42 781 394 859 643 643 0 571 357 16 689 823 4 328 311 716 464 209 863 13 13 0 223 549 234 434 827 732 727 373 463 629 574 863 169 169 0 620 472 597 754 378 551 481 279 579 638 756 863 434 434 0 362 749 6 860 254 560 395 438 178 791 680 877 166 166 0 82 875 839 55 93 562 657 372 541 830 526 883 644 644 0 798 613 531 182 277 738 635 393 228 723 250 883 657 657 0 539 735 114 572 515 757 846 839 13 214 216 883 754 754 0 765 92 747 821 54 410 403 875 655 14 828 887 292 292 0 455 170 748 775 370 490 186 715 498 157 698 907 127 127 0 137 271 198 33 837 240 191 571 828 476 309 907 761 761 0 455 797 276 341 797 449 578 290 269 406 425 907 771 771 0 409 815 892 441 213 535 209 336 728 505 730 911 328 328 0 672 30 58 222 866 865 17 817 391 875 619 919 457 457 0 855 700 794 688 724 363 145 798 323 119 552 941 292 292 0 172 395 299 939 178 489 440 7 689 635 512 947 208 208 0 312 35 913 584 565 6 410 775 879 299 809 953 325 325 0 572 927 616 271 312 900 736 196 723 30 312 967 287 287 0 825 375 427 750 261 32 728 516 219 280 682 971 296 296 0 948 665 509 708 761 603 285 562 329 661 4 977 238 238 0 347 362 124 792 240 200 756 161 243 741 462 977 354 354 0 850 276 686 637 400 930 929 95 105 884 144 983 419 419 0 213 574 546 313 131 413 53 233 388 969 833 991 976 976 0 575 10 287 151 223 170 673 938 644 102 782 997 252 252 0 263 89 35 122 232 177 340 31 656 430 975 1009 990 990 0 124 622 369 650 10 652 877 737 519 691 289 1013 899 899 0 53 939 233 19 435 933 764 290 801 955 370 1019 705 705 0 567 21 532 557 518 854 25 529 594 417 158 1021 551 551 0 260 41 772 221 55 674 42 959 304 485 635 1021 637 637 0 589 121 633 88 773 342 354 91 804 655 333 1031 158 158 0 480 830 356 934 477 622 710 463 770 100 369 1051 806 806 0 473 472 90 578 635 631 429 987 224 39 242 1061 508 508 0 688 141 676 860 1001 444 150 103 850 825 635 1061 917 917 0 927 579 89 823 291 615 182 526 837 78 720 1069 487 487 0 200 281 102 91 1020 732 285 585 688 933 294 1087 617 617 0 69 32 215 116 253 343 137 787 598 20 1069 1087 740 740 0 667 801 656 635 137 1075 554 970 612 603 320 1091 691 691 0 344 654 419 729 142 357 405 134 588 560 604 1093 2 2 0 974 227 564 478 1009 395 581 1027 834 136 534 1093 4 4 0 617 911 22 127 436 652 1082 282 490 680 617 1093 8 8 0 758 548 1017 484 587 858 153 261 275 1033 699 1093 16 16 0 564 734 507 841 302 601 527 806 750 801 954 1093 32 32 0 317 743 855 245 337 933 967 356 787 38 234 1093 64 64 0 105 35 624 721 693 233 641 256 810 308 709 1093 128 128 0 245 446 816 460 1024 275 545 532 401 285 940 1093 256 256 0 560 551 976 443 773 450 275 240 1071 524 75 1093 512 512 0 167 421 1054 720 588 456 112 1072 227 128 1009 1093 1024 1024 0 614 571 46 499 179 113 286 721 989 468 139 1097 425 425 0 26 245 1049 249 341 724 106 41 347 69 1011 1097 579 579 0 763 103 818 258 393 810 261 805 264 120 973 1097 776 776 0 620 916 30 815 652 151 357 698 307 129 138 1103 284 284 0 835 17 453 446 84 354 412 589 730 1074 886 1103 793 793 0 580 1071 306 463 830 699 859 172 222 268 395 1103 1054 1054 0 344 685 433 201 756 483 5 245 675 558 634 1109 76 76 0 939 809 237 480 479 1107 637 129 392 867 156 1109 1082 1082 0 783 361 697 958 48 616 149 794 60 133 1028 1117 1066 1066 0 898 390 571 219 1096 267 878 759 80 868 135 1123 897 897 0 639 1027 895 574 845 571 509 433 73 773 113 1123 1012 1012 0 371 669 175 423 392 604 1042 815 547 645 614 1163 78 78 0 1098 801 411 910 227 499 290 721 643 885 799 1163 170 170 0 428 1101 877 826 57 692 90 574 687 744 605 1163 241 241 0 364 519 987 983 184 662 156 159 92 997 997 1163 618 618 0 941 548 1152 1052 932 236 827 244 465 173 961 1181 874 874 0 304 948 177 215 662 90 55 13 207 287 575 1187 184 184 0 350 618 719 797 67 157 998 345 1017 536 1036 1187 315 315 0 503 1000 945 140 370 541 378 988 948 1132 40 1193 622 622 0 941 639 206 110 543 439 449 389 884 88 140 1201 206 206 0 411 1159 1118 848 842 465 457 1129 292 91 517 1213 178 178 0 20 837 172 364 671 1058 94 1145 1170 110 1039 1217 1188 1188 0 126 385 746 59 346 177 424 356 890 72 454 1223 997 997 0 743 770 969 557 48 55 398 223 771 332 183 1229 821 821 0 1021 765 663 1085 785 661 1217 812 285 612 354 1249 326 326 0 385 232 1234 1185 1083 219 1074 176 99 406 24 1279 683 683 0 170 53 846 1261 481 971 418 535 375 587 287 1283 45 45 0 791 1182 60 228 365 911 1265 1000 455 251 524 1291 62 62 0 17 698 644 798 311 167 837 823 866 1160 319 1291 1148 1148 0 143 119 24 605 147 961 971 1103 732 503 1076 1291 1286 1286 0 1166 961 41 104 765 57 690 797 190 757 476 1297 156 156 0 200 204 688 901 932 41 624 1128 67 85 477 1303 528 528 0 383 754 864 147 1292 651 1285 338 1055 152 1249 1321 403 403 0 899 701 737 674 388 440 885 1064 1095 252 1307 1327 585 585 0 968 642 607 640 378 1080 438 1073 247 1241 448 1327 1149 1149 0 707 865 1142 291 1174 216 1326 266 54 448 133 1361 228 228 0 1135 1151 901 1336 386 44 558 1234 923 1313 838 1367 411 411 0 821 461 915 1129 35 538 716 634 223 1124 1358 1373 884 884 0 291 855 67 378 133 350 331 244 979 1343 1014 1381 429 429 0 250 1342 1333 1196 842 241 273 1002 71 659 223 1381 653 653 0 1157 633 198 1002 1069 345 416 171 120 1354 990 1399 328 328 0 234 1124 836 567 1292 70 462 1223 28 412 264 1399 987 987 0 1081 1095 1373 1158 944 999 786 764 1083 941 1130 1409 1164 1164 0 191 900 260 426 1401 1126 1215 748 1271 31 1220 1429 688 688 0 965 39 739 1081 280 995 462 636 820 913 1174 1429 1401 1401 0 1261 620 1297 515 194 1253 144 945 639 703 502 1439 319 319 0 180 1292 1262 213 1217 1281 930 1111 698 69 362 1447 584 584 0 784 664 929 1198 1312 882 256 972 75 137 341 1453 365 365 0 1325 1105 1307 1371 702 255 1159 856 945 424 960 1453 378 378 0 1242 1441 129 59 662 946 1278 1063 801 1452 117 1459 1082 1082 0 544 379 742 1112 1340 292 1061 1180 1124 698 876 1471 1195 1195 0 1397 299 935 820 319 95 1171 855 395 152 920 1483 421 421 0 1302 215 604 640 1343 1203 990 1062 138 975 162 1483 1061 1061 0 264 408 907 1421 487 226 897 1365 1434 749 1481 1489 1211 1211 0 556 88 270 1432 1470 412 126 518 949 6 582 1493 164 164 0 704 518 340 810 833 1474 846 704 592 817 152 1493 488 488 0 1016 1446 1266 958 950 854 799 1103 245 633 690 1499 941 941 0 981 1381 1464 692 260 963 1104 557 22 1084 634 1499 1172 1172 0 1375 855 1059 635 118 1181 1408 469 1232 1096 1 1511 934 934 0 953 1124 161 281 787 1217 569 997 23 524 976 1523 1032 1032 0 1478 524 1315 132 326 23 62 1499 1183 1429 309 1523 1246 1246 0 646 83 120 916 960 240 680 604 78 111 1322 1531 472 472 0 574 1110 1386 467 1395 180 765 1143 1155 164 1241 1531 1238 1238 0 1323 1331 401 904 724 1315 853 926 280 731 835 1549 855 855 0 1003 313 1545 254 1010 256 1073 197 1028 586 448 1549 1069 1069 0 1248 215 543 885 1211 194 1459 585 265 1319 444 1553 392 392 0 588 1267 615 1068 1300 144 1149 1018 230 347 52 1553 568 568 0 321 774 1018 641 1413 1423 1035 1369 225 199 875 1559 1454 1454 0 56 668 1379 798 1209 98 538 701 963 826 19 1571 1265 1265 0 116 859 989 111 649 1 538 780 6 1080 201 1579 603 603 0 709 1177 1270 1286 1092 1305 592 55 1000 1327 1496 1579 906 906 0 201 345 1225 73 749 532 134 1381 1115 754 24 1579 1235 1235 0 619 916 1548 1353 1051 681 1411 487 199 1283 289 1597 453 453 0 473 345 564 567 998 205 1548 1152 1016 148 499 1601 1420 1420 0 1 1562 600 336 214 1199 81 638 1296 384 139 1607 874 874 0 669 1209 508 82 402 146 1188 450 643 441 92 1607 1253 1253 0 64 1021 489 1135 786 892 1418 1564 1437 326 1497 1613 35 35 0 359 1194 748 952 514 222 858 510 851 401 1067 1613 1225 1225 0 935 1332 636 1408 745 906 434 185 333 645 329 1619 371 371 0 37 1339 439 801 1251 1424 19 782 729 1067 226 1619 536 536 0 22 116 95 1380 1340 394 1364 1198 1129 710 24 1621 558 558 0 256 278 405 1138 1452 564 925 313 1599 732 636 1621 746 746 0 1560 1116 604 652 1436 1356 39 1370 1253 486 1301 1627 923 923 0 400 1481 342 1080 1053 842 1070 1303 1240 146 1598 1657 427 427 0 1511 620 1647 1265 1649 510 96 739 1301 333 648 1657 1481 1481 0 1616 369 1222 183 464 566 254 292 475 739 177 1663 709 709 0 1426 1234 998 320 1239 271 173 352 16 68 1158 1667 463 463 0 59 1534 373 1039 315 1371 261 968 884 1362 295 1669 221 221 0