Diophantine Equation #diophantine #equation

 

 Diophantine Equation :        

A Diophantine equation is a polynomial equation in one or more variables where only integer solutions are sought.

General form:

$$f(x_1, x_2, \dots, x_n) = 0$$

where:

• $f$ has integer coefficients

• Solutions must be integers (sometimes non-negative integers)

📌 Named after Diophantus of Alexandria (3rd century).

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2. Types of Diophantine Equations

2.1 Linear Diophantine Equations

Form:

$$ax + by = c$$

where $a, b, c \in \mathbb{Z}$

Condition for solutions:

$$\gcd(a, b) \mid c$$

General solution:

If $d = \gcd(a, b)$, and $(x_0, y_0)$ is one solution, then:

$$x = x_0 + \frac{b}{d}t,\quad y = y_0 - \frac{a}{d}t,\quad t \in \mathbb{Z}$$

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2.2 Homogeneous Linear Diophantine Equations

Form:

$$ax + by = 0$$

General solution:

$$x = \frac{b}{d}t,\quad y = -\frac{a}{d}t$$

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2.3 Diophantine Equations in Three or More Variables

Example:

$$ax + by + cz = d$$

Solutions exist if:

$$\gcd(a, b, c) \mid d$$

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3. Nonlinear Diophantine Equations

3.1 Quadratic Diophantine Equations

Example:

$$ax^2 + bx + c = y^2$$

Special cases include:

• Pell’s equation

• Elliptic equations

• Hyperbolic equations

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3.2 Pell’s Equation

Form:

$$x^2 - Dy^2 = 1$$

where $D$ is a positive non-square integer.

✔ Has infinitely many integer solutions

Solutions obtained using continued fractions.

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3.3 Pythagorean Equation

$$x^2 + y^2 = z^2$$

Primitive solutions:

$$x = m^2 - n^2,\quad y = 2mn,\quad z = m^2 + n^2$$

where $m > n$, $\gcd(m, n) = 1$, one even.

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4. Exponential Diophantine Equations

Variables appear as exponents.

Example:

$$2^x + 3^y = 5^z$$

⚠ Very difficult in general.

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4.1 Fermat’s Last Theorem

$$x^n + y^n = z^n,\quad n > 2$$

No non-zero integer solutions.

✔ Proved by Andrew Wiles (1994).

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5. Methods of Solving Diophantine Equations

5.1 Euclidean Algorithm

Used for linear equations.

5.2 Modular Arithmetic

Reduce equation modulo $n$ to show impossibility.

5.3 Infinite Descent

Used by Fermat.

5.4 Continued Fractions

Used in Pell’s equation.

5.5 Factorization

Example:

$$x^2 - y^2 = n \Rightarrow (x+y)(x-y) = n$$

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6. Diophantine Approximation

Study of approximating real numbers by rationals.

Example:

$$\left|x - \frac{p}{q}\right| < \frac{1}{q^2}$$

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7. Hilbert’s Tenth Problem

Asked for an algorithm to solve all Diophantine equations.

❌ No general algorithm exists
(Proved by Matiyasevich, 1970)

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8. Important Theorems

Theorem	Statement
Bézout’s Identity	$ax + by = \gcd(a,b)$
Fermat’s Last Theorem	No solutions for $n>2$
Lagrange’s Four Square Theorem	Every integer is sum of 4 squares
Mordell’s Theorem	Elliptic equations have finitely many solutions

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9. Applications

• Cryptography (RSA, elliptic curves)

• Coding theory

• Computer science (logic, undecidability)

• Number theory research

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10. Example Problems

Example 1:

Solve $15x + 21y = 3$

$$\gcd(15,21) = 3 \Rightarrow \text{solutions exist}$$

One solution: $x = -4, y = 3$

General solution:

$$x = -4 + 7t,\quad y = 3 - 5t$$

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Example 2:

Solve $x^2 - 5y^2 = 1$

Smallest solution: $(9,4)$
Infinitely many solutions follow.