Quadratic diophantine equations BCMATH programs
- Nagell fundamental solutions:
solving x2 – dy2 = n, d > 0, n nonzero:
- Stolt fundamental solutions:
solving x2 – dy2 = 4n, d > 0, n nonzero:
- Dujella's unicity conjecture
- Solving ax2 + bxy + cy2 = n, where d = b2 – 4ac > 0, d non-square, gcd(a,b,c)=1
- 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)
- Solving the quadratic diophantine equation ax2 + bxy + cy2 + dx + ey + f = 0.
- 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
- 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 15th May 2017
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