In the example, we saw that \(\varphi^2=\varphi+1\). Can you write \(\varphi^3\) in the form \(a\varphi+b\), where \(a\) and \(b\) are positive integers?
Let \(m\) and \(n\) be positive integers. What positive integers can be written as \(m+n+\gcd(m,n)+\text{lcm}(m,n)\), for some \(m\) and \(n\)?
Let \(n\ge r\) be positive integers. What is \(F_n^2-F_{n-r}F_{n+r}\) in terms of \(F_r\)?
On the questioners’ planet (where everyone can only ask questions. Cricks can only ask questions to which the answer is yes, and Goops can only ask questions to which the answer is no), you meet 4 alien mathematicians.
They’re called Alexander Grothendieck, Nicolas Bourbaki, Henri Cartan and Daniel Kan (you may like to shorten their names to \(A\), \(B\), \(C\) and \(D\)).
Alexander asks the following question “Am I the kind who could ask whether Bourbaki could ask whether Cartan could ask whether Daniel is a Goop?"
Amongst the final three (that is, Bourbaki, Cartan and Daniel), are there an even or an odd number of Goops?
Four different digits are given. We use each of them exactly once to construct the largest possible four-digit number. We also use each of them exactly once to construct the smallest possible four-digit number which does not start with \(0\). If the sum of these two numbers is \(10477\), what are the given digits?
Have you wondered if \(F_{-5}\) is possible? Here is how we can extend the Fibonacci sequence to the negative indices. The relation \(F_{n+1} = F_n + F_{n-1}\) can be rewritten as \(F_{n-1} = F_{n+1} - F_n\). We can simply define the Fibonacci sequence with negative indices with this formula. For example, \(F_{-1} = F_1 - F_0 = 1 - 0 = 1\).
Write out \(F_{-1}, F_{-2},\dots,F_{-10}\). What do you notice about the Fibonacci sequence with negative indices?
There are \(19\) aliens on the planet of Cricks and Goops standing in a circle. Each of them asks the following question “Do I have a Crick standing next to me on both sides?" Then one of them asks you in private “Is \(57\) a prime number?" How many Cricks were actually in the circle?
Recall that these aliens can only ask questions. Cricks can only ask questions to which the answer is yes, Goops can only ask questions to which the answer is no.