3.0 Number Theory

Let \( F_n = 2^{2^n} + 1 \) be the \( n \)-th Fermat number for \( n \ge 0 \).
a) Prove by induction that for any \( n \ge 1 \), \( F_n = F_0 \cdot F_1 \cdot \dots \cdot F_{n-1} + 2 \).
b) Use the result from part (a) to show that any two distinct Fermat numbers \( F_m \) and \( F_n \) (where \( m \ne n \)) are relatively prime, i.e., \( \gcd(F_m, F_n) = 1 \).

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