3 ms·
Examples of mathematical patterns that eventually fail
- ColinWright 5y agoMy favourite, which I think is easily explained, is this. We are mostly familiar with the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ... Each term is the sum of the two previous. Let's modify this, and instead of adding the immediate two previous terms, skip over one: K = 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, ... <- The sequence ===================================================== N = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... <- The index * * * * * * <- Does it divide? I've put the index number below, and marked those places where the index number divides into the sequence number. So as an example, the 13th element is 39, and 13 divides into 39. It appears to give the primes. So we can state this more technically as: let k(1) = 0 k(2) = 2 k(3) = 3 k(n) = k(n-2)+k(n-3) for n > 3 Claim: n>1 divides k(n) if and only if n is prime.