Introduction to Integer Solutions in A2 B2 10C2 D2

The problem of finding positive integer solutions to the equation A2 B2 10C2 D2 can be explored using concepts from complex numbers, specifically Gaussian integers. This article delves into how the norms of Gaussian integers can be utilized to construct such solutions. Additionally, it presents several techniques and examples to illustrate various scenarios.

Theoretical Background

The norm of a complex number is defined as the product of the number with its complex conjugate. For a Gaussian integer u iv, where i is the imaginary unit, the norm is given by ((u iv)(u - iv) u2 v2). The norm is also multiplicative, meaning that for Gaussian integers (g) and (h), the norm of (gh) is the product of the norms of (g) and (h).

Practical Application: Gaussian Integers

Given the equation (A2 B2 10C2 D2), we recognize that 10 can be expressed as the norm of Gaussian integers (3 pm i). Consequently, (C2 D2) is the norm of (C pm Di). Using these facts, we can generate solutions.

Constructing Solutions

For a general integer pair ((c, d)), we have the solutions: (c di, 3 i) leading to (3c - d i) and (c 3d) (c di, 3 - i) leading to (3c d i) and (-3d c) These pairs provide the identities:

( (3c - d)2 3d2 10c2 d2 )

( (3c d)2 3d2 10c2 d2 )

Examples of these solutions include:

(c 1, d 1) giving (2^2 4^2 20) (c 2, d 1) giving (5^2 5^2 50) (c 3, d 1) giving (8^2 6^2 100) (c 3, d 2) giving (7^2 9^2 130)

Generalizing the Solutions

We can generalize the above solutions by considering (D Ck), (A 2C - k), and finding (B) such that (B2 10C2k2 - 2C - k2). This leads to the following boxed solutions:

( boxed{D Ck, A 2C - k, B 4Ck} )

(A 2C - 1, B 4C3, D C1): Example ([17, 12, 31, 112, 351, 534, 131]) (A 2C3, B 4C1, D C1): Example ([55, 127, 92, 313, 334, 113, 3391]) (A 2C - 1, B 4C6, D C2) (A 2C3, B 22C1, D C2)

More general solutions are possible by taking (A 3B) and finding (B, C, D) as follows:

(B m2n2) (C 2mn) (D m2 - n2)

For example:

([A, B, C, D] [155, 43, 351, 1125]) ([A, B, C, D] [515, 43, 512, 3135]) ([A, B, C, D] [515, 34, 511, 2135]) ([A, B, C, D] [1339, 125, 121, 3139])

These solutions demonstrate the flexibility and variety of the integer solutions.

Inferring Solutions from Pythagorean Triples

A different approach is to use the properties of Pythagorean triples. For instance, if (C equiv 0, 6, 5 pmod{10}) and (D equiv 0, 6, 5 pmod{10}), and their sum is divisible by 10, the quotient can be broken down into the sum of two squares. Examples include:

(frac{8^2 6^2}{10} 10 1^2 3^2); solutions: ([8, 6, 1, 3]), ([8, 6, 3, 1]), ([6, 8, 1, 3]), ([6, 8, 3, 1]) (frac{5^2 5^2}{10} 5 1^2 2^2); solutions: ([5, 5, 1, 2]), ([5, 5, 2, 1]) (frac{10^2 10^2}{10} 20 4^2 2^2); solutions: ([10, 10, 4, 2]), ([10, 10, 2, 4]) (frac{4^2 2^2}{10} 2 1^2 1^2); solutions: ([4, 2, 1, 1]), ([2, 4, 1, 1])

In conclusion, the application of Gaussian integers and the properties of Pythagorean triples offer powerful methods for finding integer solutions to complex equations. The flexibility in constructing such solutions provides a rich field for exploration and practical application.