Fibonacci! 1,1,2,3,5,8,...
You can do a lot of cool things with the Fibonacci sequence. So many, in fact, that people have written whole books on them. Obviously then, I don't have time for them all here. However, here are just a couple things that I found out about it today!
Let the nth Fibonacci number be an.
1) If you add an and an+3, the sum will be divisible by 2. Similarly, the sum of an and an+4 is divisible by 3, and the sum of an and an+8 is divisible by 7. Let's call this property P(k). So P(2) works, P(3) works, P(7) works. But this doesn't work for some numbers. As examples, P(4) and P(11) don't work at all, and instead of an and an+6, instead it's an and an+10 for 5. Interesting.
2) Any 2 consecutive Fibonacci numbers an and an+1, n > 2, are relatively prime! This is somewhat surprising but even more surprisingly easy to prove if you write them in terms of the Euclidean Algorithm. :)
You can do a lot of cool things with the Fibonacci sequence. So many, in fact, that people have written whole books on them. Obviously then, I don't have time for them all here. However, here are just a couple things that I found out about it today!
Let the nth Fibonacci number be an.
1) If you add an and an+3, the sum will be divisible by 2. Similarly, the sum of an and an+4 is divisible by 3, and the sum of an and an+8 is divisible by 7. Let's call this property P(k). So P(2) works, P(3) works, P(7) works. But this doesn't work for some numbers. As examples, P(4) and P(11) don't work at all, and instead of an and an+6, instead it's an and an+10 for 5. Interesting.
2) Any 2 consecutive Fibonacci numbers an and an+1, n > 2, are relatively prime! This is somewhat surprising but even more surprisingly easy to prove if you write them in terms of the Euclidean Algorithm. :)
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