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# Hard Prob

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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:

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

$$\ldots$$

Oct 22, 2019