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:

  1. Are there any solutions?
  2. Are there any solutions beyond some that are easily found by inspection?
  3. Are there finitely or infinitely many solutions?
  4. Can all solutions be found in theory?
  5. 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 \(n\):

  1. Reduction of Variables – Here we select n such that coefficients vanish and result in a better computational form.
  2. 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.

For detailed information visit here.

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.

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