Here we shall aim to generalise the corollary from the last post, and on the way generalise some other important properties of the Fibonacci sequence, to general Fibonacci-like sequences.

Consider a sequence with and for , where such that . Let be the roots of . Then

.

Let be an odd prime. Suppose and have a common factor. By Dedekind’s factorisation criterion, is either prime or it factors into two (not necessarily distinct) prime ideals each with norm . In the first case, we must have i.e. , taking ideal norms. In the second case, if one of the prime factors of divides , then (again taking norms) must divide .

So we take which ensures that and are coprime. Thus we deduce from the general exponent lifting result,

**Lemma 1. **If is a prime factor of such that , then .

Next, recalling the proof of the corollary from last time, we seem to require the condition .

**Lemma 2. **If , then for all .

*Proof. *First, it is clear that , where , and that . Thus .

Let be a prime factor of . Then for ,

.

If and share a common factor, then (taking ideal norms,) . Hence , a contradiction as .

Hence and by the Euclidean algorithm we get . Thus , as desired.

Now we define to be the least such that , and we are ready for our generalisation!

**Theorem. **Let be given by and for , where are coprime integers such that . Then for an odd prime for which exists,

*Proof. *If , then . By definition of , . Hence , which is impossible as . Thus and the proof of the corollary in the previous post now applies.

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