Quadratic diophantine equations BCMATH programs

• Nagell fundamental solutions: solving x² – dy² = n, d > 0, n nonzero:
  • for fundamental solutions, by the Lagrange-Mollin-Matthews method;
  • for least positive primitive solutions, by the Lagrange-Mollin-Matthews method;
  • for fundamental solutions, by Trygve Nagell's method;
• Stolt fundamental solutions: solving x² – dy² = 4n, d > 0, n nonzero:
  • for fundamental solutions, by Bengt Stolt's method.
  • for the Stolt fundamental solutions using LMM.
• Dujella's unicity conjecture
  • Calculating a continued fraction arising from an exceptional solution to Dujella's diophantine equation x² – (k² + 1)y² = k².
  • Testing Dujella's unicity conjecture.
  • Testing Dujella's unicity conjecture using the equation X² – (k² + 1)Y² = -k².
• Solving ax² + bxy + cy² = n, where d = b² – 4ac > 0, d non-square, gcd(a,b,c)=1
  • for primitive solution classes, by the Lagrange-Matthews method.
  • for primitive and imprimitive solution classes, by the Lagrange-Matthews method.
  • for fundamental solutions, by Bengt Stolt's method.
  • using completion of the square, together with the Stolt fundamental solutions of the reduced equation X² – D y² = 4an, where X=2ax + by.
  • when |n| < (√d)/2.
• Solving ax² – by² = c, a > 0, b > 0, c ≠ 0, ab not a perfect square, using the LMM method.
• Solving the diophantine equation ax² + bxy + cy² = m, (d = b² – 4ac < 0, a > 0, m > 0)
  • using the algorithm in Dickson's Introduction to the theory of numbers for finding the primitive solutions;
  • using a variant of the algorithm in Dickson's Introduction to the theory of numbers to find
    • the primitive solutions;
    • the primitive and imprimitive solutions;
  • using a continued fraction method of Lagrange for finding the primitive solutions.
• Solving the quadratic diophantine equation ax² + bxy + cy² + dx + ey + f = 0 (general case).
• Solving the quadratic diophantine equation ax² + bxy + cy² + dx + ey + f = 0 when b² – 4ac > 0 is nonsquare
  • Legendre transformation.
  • Florida transformation.
  • Lagrange transformation.
• Solving the generalized Pell equation ax² – by² = ±1.
• Finding integers x and y which give small multiples k in x² – dy² = kn, d > 0.
• Solving the Pell equation x² – dy² = ±1
  • by the nearest integer continued fraction method (NICF-H) not using midpoint criteria;
  • by the nearest integer continued fraction method (NICF-P), using midpoint criteria;
  • by the nearest square continued fraction method (NSCF) not using midpoint criteria;
  • by the nearest square continued fraction method (NSCF), using midpoint criteria;
• Solving the diophantine equation x² – xy – ¼(d – 1)y² = ±1 using the nearest square continued fraction of ½(1 + √d), d ≡ 1(mod 4).
• Solving the Pell equations x² – dy² = ±1, ±2, ±3 and ±4.
• The Diophantine equations x² – dy² = 1 and x² – dy² = 4.
• Primitive Pythagorean triples and the construction of non-square d such that the negative Pell equation x² – dy² = -1 is soluble.
• Finding the fundamental unit of a real quadratic field.
• Expressing a prime p = 4n + 1 as a sum of two squares.
• Expressing a prime 5n ± 1 in the form x² + xy – y², with x > y ≥ 1. (Christina Doran, Shen Lu and Barry R. Smith)
• A generalization of the Hermite-Serret algorithm.
• Solving the diophantine equation ax² + by² = m using Cornacchia's method.
• Finding all ordered Markoff triples (x,y,z) with 5 ≤ z ≤ upper bound.
• Solving Pell's equation without irrational numbers (Norman Wildberger's algorithm)

Last modified 6th February 2025
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