Problems

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 \(n\)?

Among the first \(20\) Fibonacci numbers: \(F_0 = 0,F_1 = 1,F_2 = 1, F_3 = 2, F_4 = 3,..., F_{20} = 6765\) find all the numbers, which have sum of digits equal to their index. For example \(F_1=1\) fits the description, however \(F_{20} = 6765\) does not, since \(6+7+6+5 \neq 20\).

Among the first \(20\) Fibonacci numbers: \(F_0 = 0,F_1 = 1,F_2 = 1, F_3 = 2, F_4 = 3,..., F_{20} = 6765\) check whether the numbers with prime index are prime.

Consider the Pascal’s triangle: it starts with \(1\), then each entry in the triangle is the sum of the two numbers above it. Prove that the diagonals of the Pascal’s triangle sum up to Fibonacci numbers.

Prove for any \(m,n\) that \(F_{m+n} = F_{m-1}F_n + F_mF_{n+1}\).

Denote by \(GCD(m,n)\) the greatest common divisor of numbers \(m,n\), namely the largest possible \(d\) which divides both \(n\) and \(m\). Prove for any \(m,n\) that \[GCD(F_n,F_m) = F_{GCD(m,n)}.\]

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?

Suppose that \(p\) is a prime number. How many numbers are there less than \(p^2\) that are relatively prime to \(p^2\)?