Output from startCalc: 26 x^2 - 91 xy - 26 y^2 - 58 x - 59 y - 71 = 0

by Dario Alejandro Alpern

First of all we must determine the gcd of all coefficients but the constant term, that is: gcd(26, -91, -26, -58, -59) = 1.

Dividing the equation by the greatest common divisor we obtain:
26 x^2 - 91 xy - 26 y^2 - 58 x - 59 y - 71 = 0

We try now to solve this equation module 9, 16 and 25.

There are solutions, so we must continue.

We want to convert this equation to one of the form:
x´^2 + B y^2 + C y + D = 0

Multiplying the equation by 104:
2704 x^2 - 9464 xy - 2704 y^2 - 6032 x - 6136 y - 7384 = 0

2704 x^2 + ( - 9464 y - 6032)x + ( - 2704 y^2 - 6136 y - 7384) = 0

To complete the square we should add and subtract:
( - 91 y - 58)^2

Then the equation converts to:
( 52 x - 91 y - 58)^2 + ( - 2704 y^2 - 6136 y - 7384) - ( 8281 y^2 + 10556 y + 3364) = 0

( 52 x - 91 y - 58)^2 + ( - 10985 y^2 - 16692 y - 10748) = 0

Now we perform the substitution:
x´ = 52 x - 91 y - 58

This gives:
x´^2 - 10985 y^2 - 16692 y - 10748 = 0

Multiplying the equation by -65:
- 65 x´^2 + 714025 y^2 + 1084980 y + 698620 = 0

- 65x´^2 +( 714025 y^2 + 1084980 y) + 698620 = 0

- 65x´^2 +((-845)^2 y^2 + 2*(-845)*(-642) y) + 698620 = 0

Adding and subtracting (-642)^2:
- 65x´^2 +((-845)^2 y^2 + 2*(-845)*(-642) y + (-642)^2) + 698620 - (-642)^2 = 0

- 65x´^2 +( - 845 y - 642)^2 + 286456 = 0

Making the substitution y´ = - 845 y - 642:
- 65 x´^2 + y´^2 + 286456 = 0

We have to find the continued fraction expansion of the roots of 1 t^2 - 91 t - 676 = 0, that is, (sqrt(10985) + 91) / 2

The continued fraction expansion is:
97+ //1, 9, 2, 25, 1, 2, 1, 1, 1, 6, 1, 5, 1, 2, 7, 7, 2, 1, 5, 1, 6, 1, 1, 1, 2, 1, 25, 2, 9, 1, 103//
where the periodic part is marked in bold (the period has 31 coefficients).

We have to find the continued fraction expansion of the roots of -65 t^2 + 1 = 0, that is, sqrt(260) /(-130)

Simplifying, sqrt(65) / (-65)

The continued fraction expansion is:
-1+ //1, 7, 16//
where the periodic part is marked in bold.

- 65 x^2 + y'^2 + 286456 = 0

Let x' = sy' - f'z, so [-(as^2 + bs + c)/f']y'^2 + (2as + b)y'z - af'z^2 = 1.

So - 65 s^2 +1 should be multiple of 286456.

This holds for s = 61031, 204259, 132645, 275873, 65727, 208955, 137341, 280569, 5887, 149115, 77501, 220729, 10583, 153811, 82197, 225425.

Let s = 61031. Replacing in the above equation:
845194 y'^2 - 7934030 y'z + 18619640 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 204259. Replacing in the above equation:
9467119 y'^2 - 26553670 y'z + 18619640 z^2 = 1
We have to find the continued fraction expansion of the roots of 9467119 t^2 - 26553670 t + 18619640 = 0, that is, (sqrt(260) + 26553670) / 18934238
Simplifying, (sqrt(65) + 13276835) / 9467119
The continued fraction expansion is:
1+ //2, 2, 16, 6, 1, 1, 15, 16//
where the periodic part is marked in bold.
Let s = 132645. Replacing in the above equation:
3992429 y'^2 - 17243850 y'z + 18619640 z^2 = 1
We have to find the continued fraction expansion of the roots of 3992429 t^2 - 17243850 t + 18619640 = 0, that is, (sqrt(260) + 17243850) / 7984858
Simplifying, (sqrt(65) + 8621925) / 3992429
The continued fraction expansion is:
2+ //6, 3, 1, 2, 1, 30, 16//
where the periodic part is marked in bold.
Y₀ = NUM(5) = 203
Z₀ = DEN(5) = 94
Since X'₀ = 132645 Y'₀ - 286456 Z₀:
X'₀ = 71
Y'₀ = 203
Y₀ = ( Y'₀ + 642)/(-845)
X₀ = ( X'₀ + 91 Y₀ + 58) / 52
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(5) = -203
Z₀ = DEN(5) = -94
Since X'₀ = 132645 Y'₀ - 286456 Z₀:
X'₀ = -71
Y'₀ = -203
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
X₀ = ( - X'₀ + 91 Y₀ + 58) / 52
It is not an integer number.
Y₀ = NUM(8) = -47653
Z₀ = DEN(8) = -22066
Since X'₀ = 132645 Y'₀ - 286456 Z₀:
X'₀ = 5911
Y'₀ = -47653
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Let s = 275873. Replacing in the above equation:
17269264 y'^2 - 35863490 y'z + 18619640 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 65727. Replacing in the above equation:
980264 y'^2 - 8544510 y'z + 18619640 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 208955. Replacing in the above equation:
9907429 y'^2 - 27164150 y'z + 18619640 z^2 = 1
We have to find the continued fraction expansion of the roots of 9907429 t^2 - 27164150 t + 18619640 = 0, that is, (sqrt(260) + 27164150) / 19814858
Simplifying, (sqrt(65) + 13582075) / 9907429
The continued fraction expansion is:
1+ //2, 1, 2, 3, 2, 3, 2, 1, 14, 16//
where the periodic part is marked in bold.
Let s = 137341. Replacing in the above equation:
4280119 y'^2 - 17854330 y'z + 18619640 z^2 = 1
We have to find the continued fraction expansion of the roots of 4280119 t^2 - 17854330 t + 18619640 = 0, that is, (sqrt(260) + 17854330) / 8560238
Simplifying, (sqrt(65) + 8927165) / 4280119
The continued fraction expansion is:
2+ //11, 1, 1, 1, 50, 1, 15, 16//
where the periodic part is marked in bold.
Y₀ = NUM(4) = 73
Z₀ = DEN(4) = 35
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = -67
Y'₀ = 73
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(7) = 60263
Z₀ = DEN(7) = 28893
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 7475
Y'₀ = 60263
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(9) = 15 547927 (8 digits)
Z₀ = DEN(9) = 7 454429 (7 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 1 928483 (7 digits)
Y'₀ = 15 547927 (8 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(11) = 4011 304903 (10 digits)
Z₀ = DEN(11) = 1923 213789 (10 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 497 541139 (9 digits)
Y'₀ = 4011 304903 (10 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(13) = 1 034901 117047 (13 digits)
Z₀ = DEN(13) = 496181 703133 (12 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 128363 685379 (12 digits)
Y'₀ = 1 034901 117047 (13 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
X₀ = ( - X'₀ + 91 Y₀ + 58) / 52
It is not an integer number.
Y₀ = NUM(15) = 267 000476 893223 (15 digits)
Z₀ = DEN(15) = 128 012956 194525 (15 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 33 117333 286643 (14 digits)
Y'₀ = 267 000476 893223 (15 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(17) = 68885 088137 334487 (17 digits)
Z₀ = DEN(17) = 33026 846516 484317 (17 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 8544 143624 268515 (16 digits)
Y'₀ = 68885 088137 334487 (17 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(19) = 17 772085 738955 404423 (20 digits)
Z₀ = DEN(19) = 8 520798 388296 759261 (19 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 2 204355 937727 990227 (19 digits)
Y'₀ = 17 772085 738955 404423 (20 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(21) = 4585 129235 562357 006647 (22 digits)
Z₀ = DEN(21) = 2198 332957 334047 405021 (22 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 568 715287 790197 210051 (21 digits)
Y'₀ = 4585 129235 562357 006647 (22 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(23) = 1 182945 570689 349152 310503 (25 digits)
Z₀ = DEN(23) = 567161 382193 795933 736157 (24 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 146726 339893 933152 202931 (24 digits)
Y'₀ = 1 182945 570689 349152 310503 (25 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(25) = 305 195372 108616 518939 103127 (27 digits)
Z₀ = DEN(25) = 146 325438 273042 016856 523485 (27 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 37 854826 977346 963071 146147 (26 digits)
Y'₀ = 305 195372 108616 518939 103127 (27 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(27) = 78739 223058 452372 537136 296263 (29 digits)
Z₀ = DEN(27) = 37751 395913 062646 553049 322973 (29 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 9766 398633 815622 539203 502995 (28 digits)
Y'₀ = 78739 223058 452372 537136 296263 (29 digits)
Y₀ = ( Y'₀ + 642)/(-845)
X₀ = ( X'₀ + 91 Y₀ + 58) / 52
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(29) = 20 314414 353708 603498 062225 332727 (32 digits)
Z₀ = DEN(29) = 9 739713 820131 889768 669868 803549 (31 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 2 519692 992697 453268 151432 626563 (31 digits)
Y'₀ = 20 314414 353708 603498 062225 332727 (32 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(31) = 5241 040164 033761 250127 516999 547303 (34 digits)
Z₀ = DEN(31) = 2512 808414 198114 497670 273101 992669 (34 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 650 071025 717309 127560 530414 150259 (33 digits)
Y'₀ = 5241 040164 033761 250127 516999 547303 (34 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(33) = 1 352168 047906 356693 929401 323657 871447 (37 digits)
Z₀ = DEN(33) = 648294 831149 293408 509161 790445 305053 (36 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 167715 804942 073057 457348 695418 140259 (36 digits)
Y'₀ = 1 352168 047906 356693 929401 323657 871447 (37 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(35) = 348 854115 319675 993272 535413 986731 286023 (39 digits)
Z₀ = DEN(35) = 167 257553 628103 501280 866071 661786 711005 (39 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 43 270027 604029 131514 868402 887466 036563 (38 digits)
Y'₀ = 348 854115 319675 993272 535413 986731 286023 (39 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(37) = 90003 009584 428499 907620 207407 253013 922487 (41 digits)
Z₀ = DEN(37) = 43151 800541 219554 037054 937326 950526 134237 (41 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 11163 499406 034573 857778 590596 270819 292995 (41 digits)
Y'₀ = 90003 009584 428499 907620 207407 253013 922487 (41 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(39) = 23 220427 618667 233300 172740 975657 290860 715623 (44 digits)
Z₀ = DEN(39) = 11 132997 282081 016838 058892 964281 573955 922141 (44 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 2 880139 576729 316026 175361 505434 983911 556147 (43 digits)
Y'₀ = 23 220427 618667 233300 172740 975657 290860 715623 (44 digits)
Y₀ = ( Y'₀ + 642)/(-845)
X₀ = ( X'₀ + 91 Y₀ + 58) / 52
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(41) = 5990 780322 606561 762944 659551 512173 789050 708247 (46 digits)
Z₀ = DEN(41) = 2872 270146 976361 124665 157329 847319 130101 778141 (46 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 743 064847 296757 500179 385489 811629 578362 192931 (45 digits)
Y'₀ = 5990 780322 606561 762944 659551 512173 789050 708247 (46 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(43) = 1 545598 102804 874267 606421 991549 165180 284222 012103 (49 digits)
Z₀ = DEN(43) = 741034 564922 619089 146772 532207 644053 992302 838237 (48 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 191707 850462 986705 730255 281009 894996 233534 220051 (48 digits)
Y'₀ = 1 545598 102804 874267 606421 991549 165180 284222 012103 (49 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(45) = 398 758319 743334 954480 693929 160133 104339 540228 414327 (51 digits)
Z₀ = DEN(45) = 191 184045 479888 748638 742648 152242 318610 884030 487005 (51 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 49 459882 354603 273320 905683 115063 097398 673466 580227 (50 digits)
Y'₀ = 398 758319 743334 954480 693929 160133 104339 540228 414327 (51 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(47) = 102878 100895 677613 381751 427301 322791 754421 094708 884263 (54 digits)
Z₀ = DEN(47) = 49324 742699 246374 529706 456450 746310 557554 087562 809053 (53 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 12760 457939 637181 530087 935988 405269 233861 520843 478515 (53 digits)
Y'₀ = 102878 100895 677613 381751 427301 322791 754421 094708 884263 (54 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(49) = 26 542151 272765 080917 537387 549812 120139 536302 894663 725527 (56 digits)
Z₀ = DEN(49) = 12 725592 432360 084739 915627 021644 395881 530343 707174 248669 (56 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 3 292148 688544 038231 489366 579325 444399 238873 704150 876643 (55 digits)
Y'₀ = 26 542151 272765 080917 537387 549812 120139 536302 894663 725527 (56 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(51) = 6847 772150 272495 199111 264236 424225 673208 611725 728532 301703 (58 digits)
Z₀ = DEN(51) = 3283 153522 806202 616523 702065 127803 391124 271122 363393 347549 (58 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 849 361601 186422 226542 726489 529976 249734 395554 150082 695379 (57 digits)
Y'₀ = 6847 772150 272495 199111 264236 424225 673208 611725 728532 301703 (58 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(53) = 1 766698 672619 030996 289788 635609 900411 567682 288935 066670 113847 (61 digits)
Z₀ = DEN(53) = 847040 883291 567914 978375 217175 951630 514180 419226 048309 418973 (60 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 219132 000957 408390 409791 944932 154546 987074 814097 017184 531139 (60 digits)
Y'₀ = 1 766698 672619 030996 289788 635609 900411 567682 288935 066670 113847 (61 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
X₀ = ( - X'₀ + 91 Y₀ + 58) / 52
It is not an integer number.
Y₀ = NUM(55) = 455 801409 763559 724547 566356 723117 881958 788821 933521 472357 070823 (63 digits)
Z₀ = DEN(55) = 218 533264 735701 715861 804282 329330 392869 267423 889198 100436 747485 (63 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 56 535206 885410 178303 499779 066006 343146 415567 641476 283526 338483 (62 digits)
Y'₀ = 455 801409 763559 724547 566356 723117 881958 788821 933521 472357 070823 (63 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(57) = 117594 997020 325789 902275 830245 928803 644955 948376 559604 801454 158487 (66 digits)
Z₀ = DEN(57) = 56380 735260 927751 124430 526465 750065 408640 481182 993883 864371 432157 (65 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 14585 864244 434868 593912 533207 084704 377228 229376 686784 132610 797475 (65 digits)
Y'₀ = 117594 997020 325789 902275 830245 928803 644955 948376 559604 801454 158487 (66 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(59) = 30 339053 429834 290235 062616 637092 908222 516675 892330 444517 302815 818823 (68 digits)
Z₀ = DEN(59) = 14 546011 164054 624088 387214 023881 187545 036374 877788 532838 907392 749021 (68 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 3 763096 439857 310687 051130 067648 787722 981736 763617 548829 930059 410067 (67 digits)
Y'₀ = 30 339053 429834 290235 062616 637092 908222 516675 892330 444517 302815 818823 (68 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(61) = 7827 358189 900226 554856 252816 539724 392605 657424 272878 125859 325027 097847 (70 digits)
Z₀ = DEN(61) = 3752 814499 590832 087052 776787 634880 636553 976077 988258 478554 242957 815261 (70 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 970 864295 618941 722390 597644 920180 147824 910856 783950 911337 822716 999811 (69 digits)
Y'₀ = 7827 358189 900226 554856 252816 539724 392605 657424 272878 125859 325027 097847 (70 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(63) = 2 019428 073940 828616 862678 164050 611800 384037 098786 510226 027188 554175 425703 (73 digits)
Z₀ = DEN(63) = 968211 594883 270623 835528 023995 775323 043380 791746 092898 934155 775723 588317 (72 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 250479 225173 247107 066087 141259 338829 351104 019313 495717 576328 330926 541171 (72 digits)
Y'₀ = 2 019428 073940 828616 862678 164050 611800 384037 098786 510226 027188 554175 425703 (73 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(65) = 521 004615 718543 882924 016110 072241 304774 688965 829495 365436 888787 652232 733527 (75 digits)
Z₀ = DEN(65) = 249 794838 665384 230117 479177 414122 398464 555690 294413 979666 533635 893727 970525 (75 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 64 622669 230402 134681 328091 847264 497792 437012 072025 111183 781371 556330 622307 (74 digits)
Y'₀ = 521 004615 718543 882924 016110 072241 304774 688965 829495 365436 888787 652232 733527 (75 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
X₀ = ( - X'₀ + 91 Y₀ + 58) / 52
It is not an integer number.
Y₀ = NUM(67) = 134417 171427 310380 965779 293720 474206 020069 369146 911017 772491 280025 721869 824263 (78 digits)
Z₀ = DEN(67) = 64446 100164 074248 099685 792244 819583 028532 324715 167060 661066 743904 806092 807133 (77 digits)
Since X'₀ = 137341 Y'₀ - 286456 Z₀:
X'₀ = 16672 398182 218577 500675 581609 452981 091619 398010 563165 189698 017533 202374 014035 (77 digits)
Y'₀ = 134417 171427 310380 965779 293720 474206 020069 369146 911017 772491 280025 721869 824263 (78 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Let s = 280569. Replacing in the above equation:
17862194 y'^2 - 36473970 y'z + 18619640 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 5887. Replacing in the above equation:
7864 y'^2 - 765310 y'z + 18619640 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 149115. Replacing in the above equation:
5045429 y'^2 - 19384950 y'z + 18619640 z^2 = 1
We have to find the continued fraction expansion of the roots of 5045429 t^2 - 19384950 t + 18619640 = 0, that is, (sqrt(260) + 19384950) / 10090858
Simplifying, (sqrt(65) + 9692475) / 5045429
The continued fraction expansion is:
1+ //1, 11, 1, 1, 1, 67, 16//
where the periodic part is marked in bold.
Y₀ = NUM(5) = -73
Z₀ = DEN(5) = -38
Since X'₀ = 149115 Y'₀ - 286456 Z₀:
X'₀ = -67
Y'₀ = -73
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(8) = -60263
Z₀ = DEN(8) = -31370
Since X'₀ = 149115 Y'₀ - 286456 Z₀:
X'₀ = 7475
Y'₀ = -60263
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(10) = -15 547927 (8 digits)
Z₀ = DEN(10) = -8 093498 (7 digits)
Since X'₀ = 149115 Y'₀ - 286456 Z₀:
X'₀ = 1 928483 (7 digits)
Y'₀ = -15 547927 (8 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(12) = -4011 304903 (10 digits)
Z₀ = DEN(12) = -2088 091114 (10 digits)
Since X'₀ = 149115 Y'₀ - 286456 Z₀:
X'₀ = 497 541139 (9 digits)
Y'₀ = -4011 304903 (10 digits)
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = NUM(14) = -1 034901 117047 (13 digits)
Z₀ = DEN(14) = -538719 413914 (12 digits)
Since X'₀ = 149115 Y'₀ - 286456 Z₀:
X'₀ = 128363 685379 (12 digits)
Y'₀ = -1 034901 117047 (13 digits)
Y₀ = ( Y'₀ + 642)/(-845)
X₀ = ( X'₀ + 91 Y₀ + 58) / 52
X₀ = 4611 818748 (10 digits)
Y₀ = 1224 735049 (10 digits)
Let s = 77501. Replacing in the above equation:
1362919 y'^2 - 10075130 y'z + 18619640 z^2 = 1
We have to find the continued fraction expansion of the roots of 1362919 t^2 - 10075130 t + 18619640 = 0, that is, (sqrt(260) + 10075130) / 2725838
Simplifying, (sqrt(65) + 5037565) / 1362919
The continued fraction expansion is:
3+ //1, 2, 3, 2, 3, 3, 1, 16//
where the periodic part is marked in bold.
Let s = 220729. Replacing in the above equation:
11055394 y'^2 - 28694770 y'z + 18619640 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 10583. Replacing in the above equation:
25414 y'^2 - 1375790 y'z + 18619640 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 153811. Replacing in the above equation:
5368219 y'^2 - 19995430 y'z + 18619640 z^2 = 1
We have to find the continued fraction expansion of the roots of 5368219 t^2 - 19995430 t + 18619640 = 0, that is, (sqrt(260) + 19995430) / 10736438
Simplifying, (sqrt(65) + 9997715) / 5368219
The continued fraction expansion is:
1+ //1, 6, 3, 1, 2, 1, 13, 1, 15, 16//
where the periodic part is marked in bold.
Y₀ = NUM(6) = 203
Z₀ = DEN(6) = 109
Since X'₀ = 153811 Y'₀ - 286456 Z₀:
X'₀ = -71
Y'₀ = 203
Y₀ = ( Y'₀ + 642)/(-845)
X₀ = ( X'₀ + 91 Y₀ + 58) / 52
and also:
X₀ = -2
Y₀ = -1
Y₀ = NUM(6) = -203
Z₀ = DEN(6) = -109
Since X'₀ = 153811 Y'₀ - 286456 Z₀:
X'₀ = 71
Y'₀ = -203
Y₀ = ( Y'₀ + 642)/(-845)
It is not an integer number.
Y₀ = ( - Y'₀ + 642)/(-845)
X₀ = ( - X'₀ + 91 Y₀ + 58) / 52
and also:
X₀ = -2
Y₀ = -1
Let s = 82197. Replacing in the above equation:
1533089 y'^2 - 10685610 y'z + 18619640 z^2 = 1
We have to find the continued fraction expansion of the roots of 1533089 t^2 - 10685610 t + 18619640 = 0, that is, (sqrt(260) + 10685610) / 3066178
Simplifying, (sqrt(65) + 5342805) / 1533089
The continued fraction expansion is:
3+ //2, 16, 6, 18, 16//
where the periodic part is marked in bold.
Let s = 225425. Replacing in the above equation:
11530804 y'^2 - 29305250 y'z + 18619640 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.

Since 2 * 2 is a divisor of the constant term (286456), the solutions should be 2 times the solutions of - 65 u^2 + v^2 + 71614 = 0. Let F be the constant term.

- 65 x^2 + y'^2 + 71614 = 0

Let u = sv - Fz, so [-(as^2 + bs + c)/F]v^2 + (2as + b)vz - aFz^2 = 1.

So - 65 s^2 +1 should be multiple of 71614.

This holds for s = 61031, 65727, 5887, 10583.

Let s = 61031. Replacing in the above equation:
3380776 v^2 - 7934030 vz + 4654910 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 65727. Replacing in the above equation:
3921056 v^2 - 8544510 vz + 4654910 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 5887. Replacing in the above equation:
31456 v^2 - 765310 vz + 4654910 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.
Let s = 10583. Replacing in the above equation:
101656 v^2 - 1375790 vz + 4654910 z^2 = 1
Since the gcd of the three coefficients is 2 there are no integer solutions.

X_n+1 = P X_n + Q Y_n + K
Y_n+1 = R X_n + S Y_n + L

In order to find the values of P, Q, R, S we have to find first an integer solution of the equation m^2 + bmn + acn^2 = m^2 - 91 mn - 676 n^2 = 1.

We have to find the continued fraction expansion of the roots of 1 t^2 - 91 t - 676 = 0, that is, (sqrt(10985) + 91) / 2

An integer solution of the equation m^2 + bmn + acn^2 = m^2 - 91 mn - 676 n^2 = 1 is:

m = -20 959202 739850 883948 245647 530545 (32 digits)
n = -214077 649120 834543 562852 106272 (30 digits)

Using the formulas:
P = m
Q = -Cn

K =	CD(P+S-2) + E(B-Bm-2ACn) 4AC-B²

R = An
S = m + Bn
L = D(B-Bm-2ACn) + AE(P+S-2)
4AC - B² + Dn

we obtain:

P = -20 959202 739850 883948 245647 530545 (32 digits) Q = -5 566018 877141 698132 634154 763072 (31 digits) K = -8 718342 976376 307924 004248 001090 (31 digits) R = -5 566018 877141 698132 634154 763072 (31 digits) S = -1 478136 669854 940484 026105 859793 (31 digits) L = -2 315281 844744 993080 179339 614324 (31 digits)

Another integer solution of the equation m^2 + bmn + acn^2 = m^2 - 91 mn - 676 n^2 = 1 is:

m = -1 478136 669854 940484 026105 859793 (31 digits)
n = 214077 649120 834543 562852 106272 (30 digits)

Using the formulas:
P = m
Q = -Cn

K =	CD(P+S-2) + E(B-Bm-2ACn) 4AC-B²

R = An
S = m + Bn
L = D(B-Bm-2ACn) + AE(P+S-2)
4AC - B² + Dn

we obtain:

P = -1 478136 669854 940484 026105 859793 (31 digits) Q = 5 566018 877141 698132 634154 763072 (31 digits) K = 3 912238 321752 930146 204026 268958 (31 digits) R = 5 566018 877141 698132 634154 763072 (31 digits) S = -20 959202 739850 883948 245647 530545 (32 digits) L = -14 731785 493753 396606 824761 778100 (32 digits)

Calculation time: 0h 0m 0s

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