- Nagell fundamental solutions:
solving x
^{2}– dy^{2}= n, d > 0, n nonzero: - Stolt fundamental solutions:
solving x
^{2}– dy^{2}= 4n, d > 0, n nonzero:- for fundamental solutions, by Bengt Stolt's method, when n is a multiple of 4.
- 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 x
^{2}– (k^{2}+ 1)y^{2}= k^{2}. - Testing Dujella's unicity conjecture.
- Testing Dujella's unicity conjecture using the equation X
^{2}– (k^{2}+ 1)Y^{2}= -k^{2}.

- Calculating a continued fraction arising from an exceptional solution to Dujella's diophantine equation x
- Solving ax
^{2}+ bxy + cy^{2}= n, where d = b^{2}– 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 X
^{2}– D y^{2}= 4an, where X=2ax + by. - when |n| < (√d)/2.

- Solving ax
^{2}– by^{2}= c, a > 0, b > 0, c ≠ 0, ab not a perfect square, using the LMM method. - Solving the diophantine equation ax
^{2}+ bxy + cy^{2}= m, (d = b^{2}– 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 - using a continued fraction method of Lagrange for finding the primitive solutions.

- using the algorithm in Dickson's
- Solving the quadratic diophantine equation ax
^{2}+ bxy + cy^{2}+ dx + ey + f = 0 (general case). - Solving the quadratic diophantine equation ax
^{2}+ bxy + cy^{2}+ dx + ey + f = 0 when b^{2}– 4ac > 0 is nonsquare - Solving the generalized Pell equation ax
^{2}– by^{2}= ±1. - Finding integers x and y which give small multiples k in x
^{2}– dy^{2}= kn, d > 0. - Solving the Pell equation x
^{2}– dy^{2}= ±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 x
^{2}– xy – ¼(d – 1)y^{2}= ±1 using the nearest square continued fraction of ½(1 + √d), d ≡ 1(mod 4). - Solving the Pell equations x
^{2}– dy^{2}= ±1, ±2, ±3 and ±4. - The Diophantine equations x
^{2}– dy^{2}= 1 and x^{2}– dy^{2}= 4. - Primitive Pythagorean triples and the construction of non-square d such that the negative Pell equation x
^{2}– dy^{2}= -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 x
^{2}+ xy – y^{2}, with x > y ≥ 1. (Christina Doran, Shen Lu and Barry R. Smith) - A generalization of the Hermite-Serret algorithm.
- Solving the diophantine equation ax
^{2}+ by^{2}= 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 26th June 2020*

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