Sum of 2 Squares in 2 Distinct Ways/Examples/65
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Theorem
$65$ can be expressed as the sum of two square numbers in two distinct ways:
\(\ds 65\) | \(=\) | \(\ds 8^2 + 1^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7^2 + 4^2\) |
Proof
We have that:
- $65 = 5 \times 13$
Both $5$ and $13$ can be expressed as the sum of two distinct square numbers:
\(\ds 5\) | \(=\) | \(\ds 1^2 + 2^2\) | ||||||||||||
\(\ds 13\) | \(=\) | \(\ds 2^2 + 3^2\) |
Thus:
\(\ds \) | \(=\) | \(\ds \paren {1^2 + 2^2} \paren {2^2 + 3^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 \times 2 + 2 \times 3}^2 + \paren {1 \times 3 - 2 \times 2}^2\) | Brahmagupta-Fibonacci Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 + 6}^2 + \paren {3 - 4}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 + 6}^2 + \paren {4 - 3}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8^2 + 1^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 64 + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 65\) |
and:
\(\ds \) | \(=\) | \(\ds \paren {1^2 + 2^2} \paren {2^2 + 3^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {1 \times 2 - 2 \times 3}^2 + \paren {1 \times 3 + 2 \times 2}^2\) | Brahmagupta-Fibonacci Identity: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 - 6}^2 + \paren {3 + 4}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {6 - 2}^2 + \paren {3 + 4}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4^2 + 7^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 16 + 49\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 65\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $50$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $65$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $50$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $65$