Introduction

Can every rational number be expressed as the sum of three squares of rational numbers? This question delves into the fascinating realm of number theory and explores the boundaries of expressing rational numbers in various forms. This article will dissect the intricacies of this mathematical query and provide insights into the conditions under which this expression is possible.

Understanding Rational and Integer Numbers

To start, we need to clarify the nature of rational and integer numbers. A rational number is any number that can be expressed as the quotient or fraction p/q, where p and q are integers and q is non-zero. In contrast, an integer is a whole number, positive, negative, or zero (e.g., -3, -1, 0, 1, 2, etc.).

The Challenge: Rational vs. Integer Numbers

The question initially posed is whether every rational number can be expressed as the sum of three squares of integers. However, this is not the case. If the rational number is negative or is not an integer, it cannot be expressed in this form. This limitation arises because the sum of squares of integers always results in a non-negative integer, which does not accommodate negative rational numbers or non-integers.

The Modified Question: Positive Rational Numbers and Rational Squares

The more interesting question is whether every positive rational number can be expressed as the sum of three squares of rational numbers. To approach this, we need to consider the properties of both rational and integer numbers.

Lagrange's Four-Square Theorem

Lagrange's Four-Square Theorem states that any positive integer can be expressed as the sum of four squares. Using this theorem, we can rewrite a positive rational number r a/b as:

[ r frac{ab}{b^2} ]

Then, according to Lagrange’s four-square theorem, any integer can be expressed as the sum of four squares. Hence, for some integers x_1, x_2, x_3, x_4:

[ ab x_1^2 x_2^2 x_3^2 x_4^2 ]

Therefore:

[ r frac{x_1^2}{b^2} frac{x_2^2}{b^2} frac{x_3^2}{b^2} frac{x_4^2}{b^2} ]

This gives us a way to express the positive rational number as a sum of four squares. However, can we reduce this to three squares?

Reducing to Three Squares

To express a rational number as the sum of three squares of rational numbers, we need to prove that it can be done under certain conditions. Let’s consider a rational number r a/b and rewrite it in a similar manner:

[ r frac{ab}{b^2} frac{x_1^2}{y_1^2} frac{x_2^2}{y_2^2} frac{x_3^2}{y_3^2} ]

By clearing denominators, we get:

[ abY^2 z_1^2 z_2^2 z_3^2 ]

where Y y_1 y_2 y_3.

On the right-hand side, we have a sum of three integer squares. If the left-hand side (i.e., abY^2) can be expressed as a sum of three integer squares, then the rational number can be expressed as the sum of three rational squares. However, not all rational numbers satisfy this condition.

Conclusion: The Role of Legendre's Three-Square Theorem

Legendre's Three-Square Theorem states that an integer can be expressed as the sum of three squares if and only if it is not of the form 4k8m7, where k and m are non-negative integers. If a rational number a/b is of the form 4k8m7, then it cannot be expressed as the sum of three integer squares, and thus, the corresponding rational number also cannot be expressed as the sum of three rational squares.

Furthermore, this theorem implies that the expression in three rational squares is possible for any non-negative integer, as every non-negative integer can be expressed as the sum of three integer squares.

Hence, while every positive non-negative integer can be expressed as the sum of three integer squares, the same is not true for all rational numbers due to specific conditions dictated by Legendre's theorem.

Therefore, the answer to the original question is dependent on the interpretation of the condition. If the three squares must be integer squares, the answer is no. If they can be squares of rational numbers, then under specific conditions, the answer is yes.