Explore Diophantine Equations

Solve Diophantine equations — polynomial w/ integer-only solutions. Classify by type, test solvability, find particular + general, generate families. Linear, Pell, Pythagorean, general quadratic.

Use When

• All integer solutions ax + by = c
• Pell's x^2 - Dy^2 = 1 (or = -1)
• Pythagorean triples or parametric families
• Prove no integer solutions (modular constraints)
• Test solvability general quadratic Diophantine
• Fundamental solution from which all others generate

In

• Required: Equation explicit form (e.g., 3x + 5y = 17 or x^2 - 7y^2 = 1)
• Optional: Find all solutions, 1 particular, or prove non-existence
• Optional: Constraints (positive integers only)
• Optional: Express general parametrically
• Optional: Proof technique (constructive, descent, modular obstruction)

Do

Step 1: Classify

Determine structure → select method.

1. Linear: ax + by = c (a, b, c integers, x, y unknown).

   • Method: Extended Euclidean.

2. Pell: x^2 - Dy^2 = 1 (or -1, or N) (D positive non-square).

   • Method: Continued fraction of sqrt(D).

3. Pythagorean: x^2 + y^2 = z^2.

   • Method: Parametric x = m^2 - n^2, y = 2mn, z = m^2 + n^2.

4. General quadratic: ax^2 + bxy + cy^2 + dx + ey + f = 0.

   • Method: Complete square, reduce to Pell or simpler, or modular constraints.

5. Higher-order/special: Fermat-type (x^n + y^n = z^n, n > 2), sum of squares, other.

   • Method: Modular obstruction, descent, known impossibility.

Record classification + method.

→ Precise classification + strategy.

If err: no standard type → try substitution/transformation to known form. x^2 + y^2 + z^2 = n via Legendre's 3-square. No reduction → modular (Step 4).

Step 2: Linear Diophantine (if linear)

Solve ax + by = c for integer x, y.

1. Compute g = gcd(a, b) via Euclidean.

2. Test: Solutions iff g | c.

   • g !| c → prove non-existence: "gcd(a, b) = g, g doesn't divide c → no integer solutions."
   • Stop.

3. Simplify: Divide by g → (a/g)x + (b/g)y = c/g, gcd(a/g, b/g) = 1.

4. Particular via extended Euclidean:

   • 1 = (a/g)*s + (b/g)*t via back-substitution.
   • Multiply c/g: (c/g) = (a/g)(sc/g) + (b/g)(tc/g).
   • Particular: x0 = s * (c/g), y0 = t * (c/g).

5. General:

   • x = x0 + (b/g)*k
   • y = y0 - (a/g)*k
   • All integers k.

6. Apply constraints (positive solutions):

   • Solve x0 + (b/g)*k > 0 + y0 - (a/g)*k > 0 for k.
   • Range valid k or state no positive.

Example (15x + 21y = 39):

gcd(15, 21) = 3. Does 3 | 39? Yes.
Simplify: 5x + 7y = 13.
Extended Euclidean: 1 = 3*5 - 2*7.
Multiply by 13: 13 = 39*5 - 26*7.
Particular: x0 = 39, y0 = -26.
General: x = 39 + 7k, y = -26 - 5k, k in Z.
Check (k=0): 5*39 + 7*(-26) = 195 - 182 = 13. Correct.

→ General family (x, y) parameterized by k + verified particular.

If err: particular wrong → re-check extended Euclidean back-sub step-by-step. Common: sign mistake. Verify: a * x0 + b * y0 = c exactly (not modulo).

Step 3: Pell (if Pell)

Solve x^2 - Dy^2 = 1 (D positive non-square).

1. Verify D non-square: D = k^2 → x^2 - k^2*y^2 = (x - ky)(x + ky) = 1 → x - ky = x + ky = ±1 → y = 0, x = ±1 (trivial). Only interesting non-square D.

2. Continued fraction sqrt(D):

   • Init: a0 = floor(sqrt(D)), m0 = 0, d0 = 1.
   • Iterate: m_{i+1} = d_i * a_i - m_i, d_{i+1} = (D - m_{i+1}^2) / d_i, a_{i+1} = floor((a0 + m_{i+1}) / d_{i+1}).
   • Until sequence a_i repeats (periodic after a0).
   • Record period r.

3. Fundamental from convergents:

   • Compute p_i / q_i.
   • p_{r-1} / q_{r-1} (end of 1st period) → fundamental:
     • r even: (x1, y1) = (p_{r-1}, q_{r-1}) solves x^2 - Dy^2 = 1.
     • r odd: (p_{r-1}, q_{r-1}) solves x^2 - Dy^2 = -1 (negative Pell). (p_{2r-1}, q_{2r-1}) solves positive.

4. Further solutions from (x1, y1):

   • Recurrence: x_{n+1} + y_{n+1} * sqrt(D) = (x1 + y1 * sqrt(D))^{n+1}.
   • Equiv: x_{n+1} = x1 * x_n + D * y1 * y_n, y_{n+1} = x1 * y_n + y1 * x_n.

5. Present fundamental + recurrence.

Fundamentals small D:

→ Fundamental (x1, y1) verified by substitution + recurrence.

If err: continued fraction no converge to period → check iteration. Period r can be large (D = 61 → r = 11, fundamental (1766319049, 226153980)). Large D → computational tools not manual.

Step 4: Modular Constraints (general quadratic/higher)

Prove no integer solutions via modular obstruction.

1. Choose modulus m (typ 2, 3, 4, 5, 7, 8, 16).

2. Enumerate residues: Compute LHS mod m for all possible var residues.

3. Check combination gives RHS mod m.

   • None works → no solution (obstruction).

4. Common obstructions:

   • Squares mod 4: n^2 = 0 or 1 (mod 4). x^2 + y^2 = c no solution if c = 3 (mod 4).
   • Squares mod 8: n^2 = 0, 1, 4 (mod 8). x^2 + y^2 + z^2 = c no solution if c = 7 (mod 8).
   • Cubes mod 9: n^3 = 0, 1, 8 (mod 9). x^3 + y^3 + z^3 = c may obstruct certain c mod 9.

5. No obstruction found → modular can't prove non-existence. Solutions may or may not → constructive or descent.

Quadratic residues ref:

→ Proof non-existence or statement no obstruction at tested moduli.

If err: modular inconclusive → infinite descent: assume solution, derive strictly smaller, repeat → contradict positivity. Classic for x^4 + y^4 = z^2 no non-trivial.

Step 5: Generate Families

Express all solutions via fundamental + integer params.

1. Linear: x = x0 + (b/g)*k, y = y0 - (a/g)*k (Step 2).

2. Pell: Recurrence Step 3 → first several:

   (x1, y1), (x2, y2), (x3, y3), ...
   

   List ≥3-5 as sanity check.

3. Pythagorean: Primitive triples from m > n > 0, gcd(m, n) = 1, m - n odd:

   • a = m^2 - n^2, b = 2mn, c = m^2 + n^2.
   • All primitives arise (up to swap a, b).

4. General: Parametric if possible. Curve genus 0 → rational parametrization. Genus ≥1 → may be finitely many (Faltings for genus ≥ 2).

5. Verify ≥3 family members by substitution.

Example (Pell, D = 2):

Fundamental: (x1, y1) = (3, 2). Check: 9 - 2*4 = 1. Correct.
(x2, y2) = (3*3 + 2*2*2, 3*2 + 2*3) = (17, 12). Check: 289 - 2*144 = 1.
(x3, y3) = (3*17 + 2*2*12, 3*12 + 2*17) = (99, 70). Check: 9801 - 2*4900 = 1.

→ Parametric/recursive all solutions + ≥3 verified.

If err: generated fail verification → fundamental or recurrence wrong. Pell → re-derive fundamental from continued fraction. Linear → re-check extended Euclidean.

Check

• Correctly classified (linear, Pell, Pythagorean, general quadratic, higher-order)
• Linear: gcd(a, b) | c checked before solve
• Extended Euclidean back-sub verified: ax0 + by0 = c exactly
• General includes all (parameterized k or recurrence)
• Pell: D verified non-square before continued fraction
• Pell: fundamental satisfies x1^2 - D*y1^2 = 1 direct computation
• Modular obstruction proofs enumerate all residues
• ≥3 family members verified substitution
• Constraints applied after general
• Non-existence claims justified by gcd or modular

Traps

• Assume gcd | c → positive solutions: General x = x0 + (b/g)*k includes negatives. Positive may not exist even solvable over integers.
• Confuse x^2 - Dy^2 = 1 vs -1: Negative Pell has solutions only when continued fraction period odd. Applying positive formula to negative target → wrong.
• Forget trivial Pell: (x, y) = (1, 0) always satisfies x^2 - Dy^2 = 1 but useless generating non-trivial. Fundamental = smallest w/ y > 0.
• Incomplete modular obstruction: Only mod 2/4 may miss higher. First few no obstruction → try 8, 9, 16, or discriminant.
• Off-by-one continued fraction period: Convergent indices tracked. Fundamental from p_{r-1}/q_{r-1} where r = period, not p_r/q_r.
• Descent w/o base case: Must show descent terminates contradiction (x = 0 contradicts x > 0). Without base → incomplete.
• Fermat's Last Thm wrong: x^n + y^n = z^n no non-trivial for n > 2 (Wiles, 1995), but not applies diff coefficients like 2x^3 + 3y^3 = z^3.

→

• analyze-prime-numbers — Factorization + gcd prereqs
• solve-modular-arithmetic — Linear congruences ax = c (mod b) equiv to linear Diophantine
• derive-theoretical-result — Formal derivation for Diophantine impossibility