# Diophantine Equation and Diophantine Analysis

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**Diophantine equations** are equations of polynomial expressions for which rational or integer solutions are sought.

**Some Examples**: Linear Diophantine Equation, Pythagoras Equation, and Pythagorean Triplets, famous Fermat’s Equation in the Fermat’s Last Theorem, Pell’s Equation, etc.

** Rational or Integral** solutions to

- \(ax + by = c\);
- \(x^2 + y^2 = z^2\);
- \(x^n + y^n = z^n\) {where \( n \in \)
**N**}.

The questions asked in **Diophantine analysis** include:

- Are there any solutions?
- Are there any solutions beyond some that are easily found by inspection?
- Are there finitely or infinitely many solutions?
- Can all solutions be found in theory?
- Can one in practice compute a full list of solutions?

#### Problem Solving Strategies

This involves techniques like Modulo Technique, Infinite Descent Technique, Lagrange’s Technique ( for Quadratic Diophantine Equations).

## Modulo Technique

This is a very strong method to show that **No Solutions exist for a particular diophantine equation**.

In this method, we see the equation reducing modulo \(n\) (how to select the \(n\) depends on intuition and experience). This results in checking a finite number of cases, which can be easily verified by hand, and in turn, these lead to contradictions.

**Example:**

\(x^2 + 4x +1 = 4y^2 \) has no solution in integers. (Hint: Take Modulo 4)

\(x^5 + 113y^5 = 137\) has no solution in integers. (Hint: Take Modulo 11)

\(n_1^4 + … + n_8^4 = 1993\) has no solution in integers. (Hint : Take Modulo 16).

\(x^3 + y^3 + z^3 = 32\) has no solution in integers. (Hint: Take Modulo 9).

**Choice of \(n\):**

**Reduction of Variables**– Here we select n such that coefficients vanish and result in a better computational form.**Complete Residue system of power is small in size**:

For Example:

- \(x^2 = 0,1\) (mod 3, 4).
- \(x^2 = 1\) (mod 8) if \(x\) is odd.
- \(x^4 = 0, 1 \) (mod 16)
- \(x^5 = -1, 0, 1 \) (mod 11)
- \(x^3 = -1, 0, 1 \) (mod 9) , etc.

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## Infinite Descent Method

This is a proof technique to show the non-existence of positive integers’ solution to Diophantine Equations, using the idea that Natural Numbers **N** follow the Well-Ordering Principle.

Starting with a solution of a given equation in positive integers, give a smaller set of solutions to the same equation, which cannot continue forever hence giving contradiction. A detailed definition can be found here.

Example:

- \(\sqrt{2}\) is irrational.
- \(p^{\frac{1}{n}}\) is irrational for prime \(p\).
- \(x^2 + y^2 = 3z^2\) has no solution in integers. (Hint: Take Modulo 3)
- \(x^3 + 9z^3 = 3y^3\) has no solution in positive integers.
**\(x^4 + y^4 = z^2\) has no solution in non-zero integers.**(Fermat)

**Problem**: Prove that \(x^(2^n) + y^(2^n) = z^(2^n)\) has no solution in non-zero integers.

**Quadratic Diophantine Equations**

Here, we discuss the general procedure to get the solution of any Diophantine Equation of the form \(AX^2 + BY^2 = CZ^2 \) given a solution of the above equation \((x,y,z)\).

Observe that the above equation is reducible to \(x^2 + PY^2 = QZ^2\). By simple arithmetic operations, we can reduce this equation to

\( (xX + PyY)^2 + P(yX – xY)^2 = (QzZ)^2 \) i.e. of the form \( x^2 + ay^2 = z^2 \).

**Algorithm to find Solutions of the form \( x^2 + ay^2 = z^2 \). **

The approach is same as that of finding solutions to Pythagorean Triplets.

- Write it as \( ay^2 = z^2 – x^2 \).
- \( ay^2 = (z-x)(z+x) \).
- Try it for \(a\) = 2 and then generalize the method.

**Problem**: Apply the Diophantine Analysis on this class of equations.

#### A Nice Observation

\(x^2 + y^2 = Az^2\), when \( A = p^2 + q^2\), then the equation can be written as \((px – qy)^2 + (px + qy)^2 = (Az)^2 \). Thus it reduces to a pythagorean triplet. Now can you comment on the Diophantine Analysis of this class of equations.

#### General Problem:

For which \((P, Q)\), \(x^2 + PY^2 = QZ^2\) has a solution, no solutions, infinitely many solutions, etc.