Sum of 2 Squares in 2 Distinct Ways/Examples/50

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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


Proof

The smallest two positive integers which can be expressed as the sum of two distinct square numbers are:

\(\ds 5\) \(=\) \(\ds 1^2 + 2^2\)
\(\ds 10\) \(=\) \(\ds 1^2 + 3^2\)


We have that:

$50 = 5 \times 10$


Thus:

\(\ds \) \(=\) \(\ds \paren {1^2 + 2^2} \paren {1^2 + 3^2}\)
\(\ds \) \(=\) \(\ds \paren {1 \times 1 + 2 \times 3}^2 + \paren {1 \times 3 - 2 \times 1}^2\) Brahmagupta-Fibonacci Identity
\(\ds \) \(=\) \(\ds \paren {1 + 6}^2 + \paren {3 - 2}^2\)
\(\ds \) \(=\) \(\ds 7^2 + 1^2\)
\(\ds \) \(=\) \(\ds 49 + 1\)
\(\ds \) \(=\) \(\ds 50\)


and:

\(\ds \) \(=\) \(\ds \paren {1^2 + 2^2} \paren {1^2 + 3^2}\)
\(\ds \) \(=\) \(\ds \paren {1 \times 1 - 2 \times 3}^2 + \paren {1 \times 3 + 2 \times 1}^2\) Brahmagupta-Fibonacci Identity: Corollary
\(\ds \) \(=\) \(\ds \paren {1 - 6}^2 + \paren {3 + 2}^2\)
\(\ds \) \(=\) \(\ds \paren {6 - 1}^2 + \paren {3 + 2}^2\)
\(\ds \) \(=\) \(\ds 5^2 + 5^2\)
\(\ds \) \(=\) \(\ds 25 + 25\)
\(\ds \) \(=\) \(\ds 50\)

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Sum_of_2_Squares_in_2_Distinct_Ways/Examples/50&oldid=382987"