Solving the Diophantine Equation: A Comprehensive Guide

The Diophantine equation represents a fascinating challenge in number theory. Consider the specific form given by:

Diophantine equation: a2 × b2 × c2 22d d2, where GCD(a, b, c) 1.

Transformation and Methodology

Let's rewrite the given Diophantine equation in a more manageable form:

a2 × b2 × c2 d2 - 22d

Step-by-Step Solution

Start by expressing d as:

d 4w 25, where w is an integer.

The equation can then be rewritten as:

a2 × b2 × c2 (4w 25)2 - 22(4w 25)

Simplifying this expression:

A2 × B2 × C2 25 × 16w2 - 88w - 625

Assumptions and Rationalization

Assume A^2 52 and 88w 625 16w2-1, leading to:

A2 B2 C2 52(120w - 49)

Finding Specific Solutions

Now, let's explore the values of w that make 120w - 49 a perfect square. Through trial and error, we can identify several values that work:

w 0, 1, 2, 4, 11, 15, 18, 23, 37, 44...

Examples of Compliant Solutions

Let's examine some examples to illustrate the process:

Example 1

w 0 a 5, b 1, c 7, d 25 LHS 52 × 12 × 72 22 × 25 25 × 1 × 49 550 625 RHS 252 625 GCD(5, 1, 7) 1

Example 2

w 1 a 5, b 3, c 13, d 29 LHS 52 × 32 × 132 22 × 29 25 × 9 × 169 638 841 RHS 292 841 GCD(5, 3, 13) 1

Example 3

w 4 a 5, b 15, c 23, d 41 LHS 52 × 152 × 232 22 × 41 25 × 225 × 529 902 1681 RHS 412 1681 GCD(5, 15, 23) 1

Further Exploration

The RHS of the equation can be further explored using different patterns, which may provide additional solutions for other values of w. This process reveals the intricate nature of solving Diophantine equations and highlights the need for systematic approaches in number theory.