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Let \(f(n)\) return the number of distinct ordered pairs of positive integers \((a, b)\) such that for each ordered pair, \(a^2 + b^2 = n\). Note that when \(a \neq b\), \((a, b)\) and \((b, a)\) are distinct. What is the smallest positive integer \(n\) for which \(f(n) = 3\)?

 Oct 22, 2019
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Let \(f(n)\) return the number of distinct ordered pairs of positive integers \((a, b)\) such that for each ordered pair,  \(a^2 + b^2 = n\).
Note that when \(a \neq b\), \((a, b)\) and \((b, a)\) are distinct.
What is the smallest positive integer \(n\) for which \(f(n) = 3\)?

 

The smallest positive integer n = 50.

 

\(f(50) = 3 : \\ \quad 1^2 + 7^2 = 50 \\ \quad 5^2 + 5^2 = 50 \\ \quad 7^2 + 1^2 = 50\)

 

the next:
\(f(200) = 3: \\ \quad 2^2 + 14^2 = 200 \\ \quad 10^2 + 10^2 = 200 \\ \quad 14^2 + 2^2 = 200 \)

 

the next:

\(f(338) = 3: \\ \quad 7^2 + 17^2 = 338 \\ \quad 13^2 + 13^2 = 338 \\ \quad 17^2 + 7^2 = 338 \)

 

the next:

\(f(450) = 3: \\ \quad 3^2 + 21^2 = 450 \\ \quad 15^2 + 15^2 = 450 \\ \quad 21^2 + 3^2 = 450\)

 

the next:

\(f(578) = 3: \\ \quad 7^2 + 23^2 = 578\\ \quad 17^2 + 17^2 = 578\\ \quad 23^2 + 7^2 = 578\)

 

the next:

\(f(800) = 3: \\ \quad 4^2 + 28^2 = 800 \\ \quad 20^2 + 20^2 = 800 \\ \quad 28^2 + 4^2 = 800\)

 

\(\ldots\)

 

laugh

 Oct 22, 2019

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