Sum of 2 Squares in 2 Distinct Ways

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Theorem

Let $m, n \in \Z_{>0}$ be distinct positive integers that can be expressed as the sum of two distinct square numbers.

Then $m n$ can be expressed as the sum of two square numbers in at least two distinct ways.


Sequence

The sequence of positive integers which can be expressed as the sum of two square numbers in two or more different ways begins:

\(\ds 50\) \(=\) \(\ds 7^2 + 1^2\) \(\ds = 5^2 + 5^2\)
\(\ds 65\) \(=\) \(\ds 8^2 + 1^2\) \(\ds = 7^2 + 4^2\)
\(\ds 85\) \(=\) \(\ds 9^2 + 2^2\) \(\ds = 7^2 + 6^2\)
\(\ds 125\) \(=\) \(\ds 11^2 + 2^2\) \(\ds = 10^2 + 5^2\)
\(\ds 130\) \(=\) \(\ds 11^2 + 3^2\) \(\ds = 9^2 + 7^2\)
\(\ds 145\) \(=\) \(\ds 12^2 + 1^2\) \(\ds = 9^2 + 8^2\)
\(\ds 170\) \(=\) \(\ds 13^2 + 1^2\) \(\ds = 11^2 + 7^2\)


Proof

Let:

$m = a^2 + b^2$
$n = c^2 + d^2$

Then:

\(\ds m n\) \(=\) \(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\)
\(\ds \) \(=\) \(\ds \paren {a c + b d}^2 + \paren {a d - b c}^2\) Brahmagupta-Fibonacci Identity
\(\ds \) \(=\) \(\ds \paren {a c - b d}^2 + \paren {a d + b c}^2\) Brahmagupta-Fibonacci Identity: Corollary

It remains to be shown that $a \ne b$ and $c \ne d$, then the four numbers:

$a c + b d, a d - b c, a c - b d, a d + b c$

are distinct.


Because $a, b, c, d > 0$, we have:

$a c + b d \ne a c - b d$
$a d + b c \ne a d - b c$

We also have:

\(\ds a c \pm b d\) \(=\) \(\ds a d \pm b c\)
\(\ds \leadstoandfrom \ \ \) \(\ds a c \mp b c - a d \pm b d\) \(=\) \(\ds 0\)
\(\ds \leadstoandfrom \ \ \) \(\ds c \paren {a \mp b} - d \paren {a \mp b}\) \(=\) \(\ds 0\)
\(\ds \leadstoandfrom \ \ \) \(\ds \paren {a \mp b} \paren {c \mp d}\) \(=\) \(\ds 0\)

Thus $a \ne b$ and $c \ne d$ implies $a c \pm b d \ne a d \pm b c$.

The case for $a c \pm b d \ne a d \mp b c$ is similar.

$\blacksquare$


Examples

$50$ as the Sum of 2 Squares

$50$ is the smallest positive integer which can be expressed as the sum of two square numbers in two distinct ways:

\(\ds 50\) \(=\) \(\ds 5^2 + 5^2\)
\(\ds \) \(=\) \(\ds 7^2 + 1^2\)


$65$ as the Sum of 2 Squares

$65$ can be expressed as the sum of two square numbers in two distinct ways:

\(\ds 65\) \(=\) \(\ds 8^2 + 1^2\)
\(\ds \) \(=\) \(\ds 7^2 + 4^2\)


$145$ as the Sum of 2 Squares

$145$ can be expressed as the sum of two square numbers in two distinct ways:

\(\ds 145\) \(=\) \(\ds 12^2 + 1^2\)
\(\ds \) \(=\) \(\ds 9^2 + 8^2\)


Sources