Let $(u_n)_{n \in \mathbb{N}}$ defined by $u_{n+2} = |u_{n+1} - u_n|$. We will denote "$U(a,b)$" the sequence writed term by term until we hit $0$. For instance, we have : $$U(4,2) : 4, 2, |2-4|=2,|2-2|=0$$ so $$U(4,2) : 4,2,2,0$$ or $$U(10,3):10,3,7,4,3,1,2,1,1,0$$

We can easily write a code (in python) that generate such sequences :

def U(a,b):
    q = [a,b]
    n = 1
    while n != 0:
        n = abs(q[-1]-q[-2])
    return q

Let $L(a,b)$ be the length of $U(a,b)$, i.e. the length of the sequence until we hit $0$. For instance, $L(10,3) = 10$ because there is 10 terms in the sequence shown above !

And let $M(a,b)$ the lowest element of the sequence that is not $0$. For exemple, we have $$U(15,4) : 15, 4, 11, 7, 4, 3, 1, 2, 1, 1, 0$$ So $M(15,4) = 1$ (note that we also have $L(15,4)=11$).

We now state two conjectures that I'd like to prove !

(1) $$\forall a,b \in \mathbb{N}, M(a,b)=M(b,a)=\gcd(a,b)$$ Exemple : $U(15,18) : 15, 18, 3, 15, 12, 3, 9, 6, 3, 3, 0 $

We see that $M(15,18)=3$ and we have $\gcd(15,18)=3$

From that conjecture, I have the deep intuition that this sequence have much to do with prime numbers, in a way that I don't know yet...

The second conjecture is :

(2) $$\forall a, b \in \mathbb{N}, c \in \mathbb{N}^*, L(ac,bc)=L(a,b) $$

For exemple $L(4,2)=4$ and $U(2,1): 2, 1, 1, 0$

so $L(2,1)=4$.

With this conjecture, I think that there is a deeper link between $U(ac,bc)$ and $U(a,b)$ than just their lengths (like maybe $U(ac,bc)=c \times U(a,b)$ or something like that...

Edit : I don't say it because it seemed obvious to me, but we also conjecture that $L(a,b)$ is finite for all integers a and b ! It'd be good to prove this conjecture too.


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