Solving Integer Solutions for the Equation √y √x √2057

The equation √Q √R √S is an interesting dual of the Pythagorean Theorem. While the Pythagorean Theorem calculates the relationship between areas of squares, our equation describes a relation between lengths that sum to a third. The figure below illustrates the relationship between the three lengths and three areas:

Understanding and Solving the Equation

To solve the equation, we can eliminate the square roots by squaring both sides, introducing extraneous roots as a result of the process:

Start with the equation: √Q √R √S. Square both sides: Q √R √S. Multiply both sides by √Q: Q 2 √QR S. Subtract S and R^2 from both sides: 4QR S - Q - R^2.

These extraneous roots provide valuable insights. If we start from ± √Q ± √R ± √S, we still arrive at the same result:

4QR S - Q - R^2

This does not imply that the sides of the squares of areas Q and R add up to the area of S. Instead, it states that two of the lengths add to the third without specifying which. The equivalence to Q R S^2 2Q^2 R^2 S^2 holds true.

Deriving the Equation

From the picture, if T is the area of a rectangle, then T √Q √R. If we express the area S of the big square as the sum of its components, we get:

S QR 2T

Substituting T into the equation, we get:

S - Q - R^2 4T^2 4QR

Given this, we need to find integer solutions for x and y, where both are zero or natural numbers. The rational form of the equation gives:

2057 - x - y^2 4xy

This equation is symmetrical, so we can also write:

x - 2057 - y^2 42057y

Exploring the Solutions

We need to find the ways 42057y can be a perfect square such that 0 ≤ y ≤ 2057. We know that:

2057 11^2 × 17

For 2057y to be a perfect square, y must be of the form 17k^2, where k is an integer. The range for k is 0 to 11, as only these values of k will give a y in the specified range. Thus, there are 12 possible pairs (x, y). Since the problem is symmetrical in x and y, we have six pairs with 0 ≤ x ≤ y.

Let's list our 12 pairs and check them for validity:

y 17k^2, x 17k^2 - 2217k 2057

For each k from 0 to 11, we get the corresponding (x, y) pairs, confirming the solution is correct.

Answer: 12 integer solutions

The problem is symmetrical in x and y, so the six pairs with 0 ≤ x ≤ y form the complete set of integer solutions.