(Pythagorean Triples) A right triangle can have sides whose lengths are all integers. The set of three integer values for the lengths of the sides of a right triangle is called a Pythagorean triple. The lengths of the three sides must satisfy the relationship that the sum of the squares of two of the sides is equal to the square of the hypotenuse. Write an application that displays a table of the Pythagorean triples for side1, side2 and the hypotenuse, all no larger than \(500 .\) Use a triple-nested for loop that tries all possibilities. This method is an example of "brute-force" computing. You'll learn in more advanced computer science courses that for many interesting problems there's no known algorithmic approach other than using sheer brute force.

The Pythagorean Theorem is a fundamental principle in geometry, particularly useful when dealing with right angles. It describes a special relationship between the three sides of a right-angled triangle. According to this theorem, for any such triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

If we name these sides 'a', 'b', and 'c', where 'c' denotes the length of the hypotenuse, the theorem is mathematically represented as:
\[a^2 + b^2 = c^2\].
In the context of Pythagorean triples, we're looking at sets of three positive integers that fit this rule. For instance, (3, 4, 5) is a classic example of such a set, often cited in math textbooks. The numbers 3 and 4 serve as sides 'a' and 'b', while 5 is the hypotenuse 'c'. When plugged into the equation, you can easily verify that \(3^2 + 4^2 = 5^2\), or \(9 + 16 = 25\), proving the relationship holds and therefore constitutes a Pythagorean Triple.

Brute-force computation refers to the simplest, most direct approach to solving a problem: systematically enumerate all possible candidates for the solution and check each candidate to see whether it satisfies the problem's statement.

A common analogy used to explain brute-force is trying every key in a large set of keys to find the one that opens a specific lock. In computational terms, a brute-force algorithm tries all available options until a satisfactory solution is found or all possibilities are exhausted. It's often considered inefficient since it might involve a significant amount of computation, especially for large sets of data, but it's also guaranteed to find a solution if one exists within the provided constraints.

Applying this to our Pythagorean triples problem means that we attempt every combination of the three sides within the given boundary, checking each against the Pythagorean Theorem equation, to determine whether it forms a valid triple. While this method consumes more computational resources, it is straightforward and ensures that no potential solution is overlooked.

Nested loops in Java programming are essentially loops within loops, allowing programmers to perform complex iteration tasks. In the context of discovering Pythagorean triples, we use three layers of nested loops to iterate through all possible combinations of side lengths up to a certain number, in this case, 500.

Here's a simplified explanation: Imagine stacking three boxes, one inside the other. The outer box represents the first loop, the next box represents the second loop, and the innermost box represents the third loop. Each loop corresponds to one of the sides of the triangle we're testing for Pythagorean validity, labeled 'side1', 'side2', and 'hypotenuse'.

The outermost loop runs through all possible values for 'side1', the middle loop does the same for 'side2', and the innermost loop checks each possible value for the 'hypotenuse'. At every level, the loop only progresses to the next value after all the iterations in the subsequent inner loop are complete. This systematic approach is essential for the brute-force method to ensure that no combination of sides is missed while searching for triples.