26 ⁢x² - 91 ⁢x⁢y - 26 ⁢y² - 58 ⁢x - 59 ⁢y - 71 ⁢ = 0

The discriminant is D = b² − 4⁢a⁢c = 10985

We apply the transformation of Legendre Dx = X + α, Dy = Y + β, and we obtain:

α = 2⁢c⁢d - b⁢e = -2353

β = 2⁢a⁢e - b⁢d = -8346

26 X² − 91 X⁢Y − 26 ⁢Y² = 5113 418635

where the right hand side equals −D (a⁢e² − b⁢e⁢d + c⁢d² + f⁢D)

Dividing both sides by 13:

2 X² − 7 X⁢Y − 2 ⁢Y² = 393 339895 (1)

We will have to solve several quadratic modular equations. To do this we have to factor the modulus and find the solution modulo the powers of the prime factors. Then we combine them by using the Chinese Remainder Theorem.

The different moduli are divisors of the right hand side, so we only have to factor it once.

393 339895 = 5 × 13³ × 61 × 587

Searching for solutions X and Y coprime.

We have to solve:2 T² − 7 T − 2 ≡ 0 (mod 393 339895 = 5 × 13³ × 61 × 587)

Solutions modulo 5: 3

There are no solutions modulo 13³, so the modular equation does not have any solution.

Let 13⁢X' = X and 13⁢Y' = Y. Searching for solutions X' and Y' coprime.

From equation (1) we obtain 2 X'² − 7 X'⁢Y' − 2 ⁢Y'² = 393 339895 / 13² = 2 327455

We have to solve:2 T² − 7 T − 2 ≡ 0 (mod 2 327455 = 5 × 13 × 61 × 587)

Solutions modulo 5: 3

Solutions modulo 13: 5

Solutions modulo 61: 47 and 48

Solutions modulo 587: 19 and 278

1. T = -95662

The transformation X' = - 95662 ⁢Y' - 2 327455 ⁢k (2) converts 2 X'² − 7 X'⁢Y' − 2 ⁢Y'² = 2 327455 to P⁢Y'² + Q⁢Y'k + R⁢ k² = 1 (3)

where: P = (a⁢T² + b⁢T + c) / n = 7864, Q = −(2⁢a⁢T + b) = 382655, R = a⁢n = 4 654910

The continued fraction expansion of (Q + D) / (2R) = (382655 + 65) / 9 309820 is:

0+ // 24, 3, 25, 2, 7, 1, 1// (4)
Solution of (3) found using the convergent Y' / (−k) = 73 / 3 of (4)

From (2): X' = -961, Y' = 73

X = -12493, Y = 949

X + α = -14846, Y + β = -7397

These numbers are not multiple of D = 10985.

X = 12493, Y = -949

X + α = 10140, Y + β = -9295

These numbers are not multiple of D = 10985.

Solution of (3) found using the convergent Y' / (−k) = 60263 / 2477 of (4)

From (2): X' = 226929, Y' = 60263

X = 2 950077, Y = 783419

X + α = 2 947724, Y + β = 775073

These numbers are not multiple of D = 10985.

X = -2 950077, Y = -783419

X + α = -2 952430, Y + β = -791765

These numbers are not multiple of D = 10985.

Solution of (3) found using the convergent Y' / (−k) = 15 547927 / 639069 of (4)

From (2): X' = 58 546721, Y' = 15 547927

X = 761 107373, Y = 202 123051

X + α = 761 105020, Y + β = 202 114705

These numbers are not multiple of D = 10985.

X = -761 107373, Y = -202 123051

X + α = -761 109726, Y + β = -202 131397

These numbers are not multiple of D = 10985.

The continued fraction expansion of (−Q + D) / (−2R) = (-382655 + 65) / (-9 309820) is:

0+ // 24, 3, 33, 1, 1, 7// (5)
Solution of (3) found using the convergent Y' / (−k) = 73 / 3 of (5)

From (2): X' = -961, Y' = 73

X = -12493, Y = 949

X + α = -14846, Y + β = -7397

These numbers are not multiple of D = 10985.

X = 12493, Y = -949

X + α = 10140, Y + β = -9295

These numbers are not multiple of D = 10985.

Solution of (3) found using the convergent Y' / (−k) = 79097 / 3251 of (5)

From (2): X' = -21009, Y' = 79097

X = -273117, Y = 1 028261

X + α = -275470, Y + β = 1 019915

These numbers are not multiple of D = 10985.

X = 273117, Y = -1 028261

X + α = 270764, Y + β = -1 036607

These numbers are not multiple of D = 10985.

2. T = -991752

The transformation X' = - 991752 ⁢Y' - 2 327455 ⁢k (6) converts 2 X'² − 7 X'⁢Y' − 2 ⁢Y'² = 2 327455 to P⁢Y'² + Q⁢Y'k + R⁢ k² = 1 (7)

where: P = (a⁢T² + b⁢T + c) / n = 845194, Q = −(2⁢a⁢T + b) = 3 967015, R = a⁢n = 4 654910

The continued fraction expansion of (Q + D) / (2R) = (3 967015 + 65) / 9 309820 is:

0+ // 2, 2, 1, 7, 1, 1, 2, 1, 1, 8, 1, 1, 7// (8)
Solution of (7) found using the convergent Y' / (−k) = 697 / 297 of (8)

From (6): X' = 2991, Y' = 697

X = 38883, Y = 9061

X + α = 36530, Y + β = 715

These numbers are not multiple of D = 10985.

X = -38883, Y = -9061

X + α = -41236, Y + β = -17407

These numbers are not multiple of D = 10985.

Solution of (7) found using the convergent Y' / (−k) = 203273 / 86617 of (8)

From (6): X' = 765439, Y' = 203273

X = 9 950707, Y = 2 642549

X + α = 9 948354, Y + β = 2 634203

These numbers are not multiple of D = 10985.

X = -9 950707, Y = -2 642549

X + α = -9 953060, Y + β = -2 650895

These numbers are not multiple of D = 10985.

The continued fraction expansion of (−Q + D) / (−2R) = (-3 967015 + 65) / (-9 309820) is:

0+ // 2, 2, 1, 7, 1, 1, 2, 1, 3, 7, 1, 1// (9)
Solution of (7) found using the convergent Y' / (−k) = 23447 / 9991 of (9)

From (6): X' = -6239, Y' = 23447

X = -81107, Y = 304811

X + α = -83460, Y + β = 296465

These numbers are not multiple of D = 10985.

X = 81107, Y = -304811

X + α = 78754, Y + β = -313157

These numbers are not multiple of D = 10985.

Solution of (7) found using the convergent Y' / (−k) = 6 050023 / 2 577975 of (9)

From (6): X' = -1 606671, Y' = 6 050023

X = -20 886723, Y = 78 650299

X + α = -20 889076, Y + β = 78 641953

These numbers are not multiple of D = 10985.

X = 20 886723, Y = -78 650299

X + α = 20 884370, Y + β = -78 658645

These numbers are not multiple of D = 10985.

3. T = -171972

The transformation X' = - 171972 ⁢Y' - 2 327455 ⁢k (10) converts 2 X'² − 7 X'⁢Y' − 2 ⁢Y'² = 2 327455 to P⁢Y'² + Q⁢Y'k + R⁢ k² = 1 (11)

where: P = (a⁢T² + b⁢T + c) / n = 25414, Q = −(2⁢a⁢T + b) = 687895, R = a⁢n = 4 654910

The continued fraction expansion of (Q + D) / (2R) = (687895 + 65) / 9 309820 is:

0+ // 13, 1, 1, 6, 1, 14, 1, 1, 7// (12)
Solution of (11) found using the convergent Y' / (−k) = 203 / 15 of (12)

From (10): X' = 1509, Y' = 203

X = 19617, Y = 2639

X + α = 17264, Y + β = -5707

These numbers are not multiple of D = 10985.

X = -19617, Y = -2639

X + α = -21970, Y + β = -10985

Dividing these numbers by D = 10985:

x = -2
y = -1

The continued fraction expansion of (−Q + D) / (−2R) = (-687895 + 65) / (-9 309820) is:

0+ // 13, 1, 1, 6, 1, 6, 2, 7, 1, 1// (13)
Solution of (11) found using the convergent Y' / (−k) = 203 / 15 of (13)

From (10): X' = 1509, Y' = 203

X = 19617, Y = 2639

X + α = 17264, Y + β = -5707

These numbers are not multiple of D = 10985.

X = -19617, Y = -2639

X + α = -21970, Y + β = -10985

Dividing these numbers by D = 10985:

x = -2
y = -1

x = -2
y = -1

4. T = -1 068062

The transformation X' = - 1 068062 ⁢Y' - 2 327455 ⁢k (14) converts 2 X'² − 7 X'⁢Y' − 2 ⁢Y'² = 2 327455 to P⁢Y'² + Q⁢Y'k + R⁢ k² = 1 (15)

where: P = (a⁢T² + b⁢T + c) / n = 980264, Q = −(2⁢a⁢T + b) = 4 272255, R = a⁢n = 4 654910

The continued fraction expansion of (Q + D) / (2R) = (4 272255 + 65) / 9 309820 is:

0+ // 2, 5, 1, 1, 2, 1, 1, 7, 1, 7, 1// (16)
Solution of (15) found using the convergent Y' / (−k) = 1253 / 575 of (16)

From (14): X' = 4939, Y' = 1253

X = 64207, Y = 16289

X + α = 61854, Y + β = 7943

These numbers are not multiple of D = 10985.

X = -64207, Y = -16289

X + α = -66560, Y + β = -24635

These numbers are not multiple of D = 10985.

Solution of (15) found using the convergent Y' / (−k) = 337397 / 154831 of (16)

From (14): X' = 1 270491, Y' = 337397

X = 16 516383, Y = 4 386161

X + α = 16 514030, Y + β = 4 377815

These numbers are not multiple of D = 10985.

X = -16 516383, Y = -4 386161

X + α = -16 518736, Y + β = -4 394507

These numbers are not multiple of D = 10985.

The continued fraction expansion of (−Q + D) / (−2R) = (-4 272255 + 65) / (-9 309820) is:

0+ // 2, 5, 1, 1, 2, 1, 1, 5, 1, 6, 1, 1, 7// (17)
Solution of (15) found using the convergent Y' / (−k) = 14123 / 6481 of (17)

From (14): X' = -3771, Y' = 14123

X = -49023, Y = 183599

X + α = -51376, Y + β = 175253

These numbers are not multiple of D = 10985.

X = 49023, Y = -183599

X + α = 46670, Y + β = -191945

These numbers are not multiple of D = 10985.

Solution of (15) found using the convergent Y' / (−k) = 3 644987 / 1 672673 of (17)

From (14): X' = -967979, Y' = 3 644987

X = -12 583727, Y = 47 384831

X + α = -12 586080, Y + β = 47 376485

These numbers are not multiple of D = 10985.

X = 12 583727, Y = -47 384831

X + α = 12 581374, Y + β = -47 393177

These numbers are not multiple of D = 10985.

Recursive solutions:

x_n+1 = P ⁢x_n + Q ⁢y_n + K
y_n+1 = R ⁢x_n + S ⁢y_n + L

where:

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)

and also:

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)

Written by Dario Alpern. Last updated on 4 November 2024.