Can Every Real Number Be Written as the Sum of Two Squares?

When discussing real numbers and their properties, one intriguing question arises: Can every real number, whether positive, negative, or irrational, be represented as the sum of two squares?

Integer Representation as the Sum of Two Squares

For integers specifically, the answer to this question is partially positive but has strict conditions. An integer can be expressed as the sum of two squares if and only if its prime decomposition does not include factors of the form ( p^n ) where ( p ) is a prime congruent to 3 modulo 4 and ( n ) is odd.

Take, for example, the number 1399489, which has a prime decomposition of ( 7^2 cdot 13^4 ). Since 7 is a prime congruent to 3 modulo 4, but is raised to an even power, 1399489 can indeed be expressed as the sum of two squares, namely ( 1092^2 455^2 ).

In contrast, consider 36125, which decomposes into ( 5^3 cdot 17^2 ). Here, although 5 is raised to an odd power, it is congruent to 1 modulo 4, allowing it to be expressed as the sum of two squares: ( 190^2 5^2 ).

However, not all integers can be expressed as such. For instance, the number 830245379, with a prime decomposition of ( 29 cdot 31^5 ), cannot be written as a sum of two squares because 31 is a prime congruent to 3 modulo 4 and is raised to an odd power.

Sum of Two Squares for Real and Rational Numbers

When we look at the broader class of real numbers, the situation changes. Every positive real number can be written as the sum of two squares in an infinite number of ways, even if these squares themselves are irrational numbers. Taking 148.37 as an example, it can be expressed as the sum of two square roots:

( 148.37 sqrt{59}^2 sqrt{89.37}^2)

Approximating the square roots, we get:

( sqrt{59} approx 7.6811 ) and ( sqrt{89.37} approx 9.4536 )

Negative Real Numbers

For negative real numbers, a slightly different approach is used. These can be written as the sum of the squares of a real and an imaginary number or even two imaginary numbers. Consider -148.37, which can be represented as:

( -148.37 sqrt{59}^2 isqrt{89.37}^2)

Here, ( sqrt{59} ) and ( sqrt{89.37} ) are real numbers, while ( i ) is the imaginary unit.

Can Every Real Number Be Written as the Sum of Two Squares?

Let's address the second part of the question: Can every real number be expressed as the sum of two rational numbers?

The answer is no. Rational numbers are closed under addition, meaning the sum of two rational numbers is always another rational number. This property makes it impossible for an irrational number to be the sum of two rational numbers. For instance, the famous irrational number, ( sqrt{2} ), cannot be expressed as the sum of two rational numbers. Similarly, ( pi ) and other irrational numbers cannot be written as the sum of two rational squares.

While we can approximate irrational numbers with rational numbers, these approximations are always finite and not exact. For example, ( frac{22}{7} ) is close to ( pi ), but it is not equal to ( pi ). This approximation principle applies generally to all irrational numbers.

Conclusion

In summary, while every positive real number can be written as the sum of two squares in an uncountably infinite number of ways, the representation does allow for irrational numbers. Negative real numbers can be expressed as the sum of two squares, either with real and imaginary components or purely imaginary components. However, it is not possible for every real number to be the sum of two rational numbers.