Sum of 2 Squares in 2 Distinct Ways/Examples/145

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Theorem

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

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


Proof

We have that:

$145 = 5 \times 29$


Both $5$ and $29$ can be expressed as the sum of two distinct square numbers:

\(\ds 5\) \(=\) \(\ds 1^2 + 2^2\)
\(\ds 29\) \(=\) \(\ds 2^2 + 5^2\)


Thus:

\(\ds \) \(=\) \(\ds \paren {1^2 + 2^2} \paren {2^2 + 5^2}\)
\(\ds \) \(=\) \(\ds \paren {1 \times 2 + 2 \times 5}^2 + \paren {1 \times 5 - 2 \times 2}^2\) Brahmagupta-Fibonacci Identity
\(\ds \) \(=\) \(\ds \paren {2 + 10}^2 + \paren {5 - 4}^2\)
\(\ds \) \(=\) \(\ds 12^2 + 1^2\)
\(\ds \) \(=\) \(\ds 144 + 1\)
\(\ds \) \(=\) \(\ds 145\)


and:

\(\ds \) \(=\) \(\ds \paren {1^2 + 2^2} \paren {2^2 + 5^2}\)
\(\ds \) \(=\) \(\ds \paren {1 \times 2 - 2 \times 5}^2 + \paren {1 \times 5 + 2 \times 2}^2\) Brahmagupta-Fibonacci Identity/Corollary
\(\ds \) \(=\) \(\ds \paren {2 - 10}^2 + \paren {5 + 4}^2\)
\(\ds \) \(=\) \(\ds \paren {8 - 2}^2 + \paren {5 + 4}^2\)
\(\ds \) \(=\) \(\ds 8^2 + 9^2\)
\(\ds \) \(=\) \(\ds 64 + 81\)
\(\ds \) \(=\) \(\ds 145\)

$\blacksquare$


Sources