Find the number of ordered pairs of positive integers $(a,b)$ such that

- $a^2 + b^2 = c$
- $a^3 + b^3 = f ( a + b)$
- $c - f = 1729$ .

The answer is 8.

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$a^{2}+ b^{2}=c \\ a^{3}+b^{3}=f(a+b)$

$c-f=1729 \\ a^{2}+b^{2}-\frac{a^{3}+b^{3}}{a+b}=1729 \\ a^{2}+b^{2}-(a^{2}+b^{2}-ab)=1729 \\ ab=1729$

1729 has 4 positive factors and 8 positive and negative factors. Answer is $\boxed{8}$