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It's not basic algebra in the sense of my high-school "Maths B" class. But it is just the kind of thing we did in "Maths C". Here a hint: use the definition o
by adrianratnapala 10y ago
It's not basic algebra in the sense of my high-school "Maths B" class. But it is just the kind of thing we did in "Maths C".
Here a hint: use the definition of φ that
1 = φ - 1/φ
And use it to prove that if
F[k] = φ^k + 1/(-φ)^k
is an integer, then
F[k+1] = φ^(k+1) + 1/(-φ)^(k+1)
Must also be an integer. Now show that this proves by induction that all F[k] are integers and note what happens to 1/(-φ)^k as k grows to infinity.
- recursive 10y agoThat seems like a clever proof, and I understand it and even admire it. But it doesn't look like that the kind of thing that's "obvious" unless you frequently work with this kind of thing.