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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $145$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $145$