Question 495491
The base cases are true.


Assume that for some n>2 that *[tex F_{n-1}] and *[tex F_n] are relatively prime, and we want to show that *[tex F_n] and *[tex F_{n+1}] are also relatively prime. By definition,


*[tex \LARGE F_{n+1} = F_n + F_{n-1}].


For any prime p satisfying *[tex F_n \equiv 0] (mod p), it is apparent that *[tex F_{n-1} \not\equiv 0] (mod p), because the nth and (n-1)th terms have no common factor other than 1. Hence, their sum *[tex F_{n+1} \not\equiv 0] (mod p) for all p|(F_n) so we are done.