Question 3
A function satisfies $f(0,0)=5^{2003}$, $f(0,n)=0$ for all $n\in\mathbb N$, and, for all integers $m\gt 0$ and $n$,
$$f(m,n)=f(m-1,n)-2\left\lfloor\frac{f(m-1,n)}2\right\rfloor+\left\lfloor\frac{f(m-1,n-1)}2\right\rfloor+\left\lfloor\frac{f(m-1,n+1)}2\right\rfloor.$$
Show that some positive integer $M$ satisfies $f(M,n)=1$ for $|n|\le(5^{2003}-1)/2$ and $f(M,n)=0$ otherwise.
Editorial clarification: the printed condition $n\in\mathbb N$ is preserved above. It omits negative initial indices and, if zero belongs to $\mathbb N$, conflicts with $f(0,0)$. The intended well-posed initial condition used for the proof is $f(0,n)=0$ for every integer $n\ne0$. The bracket notation in the source is read as floor.
$$f(m,n)=f(m-1,n)-2\left\lfloor\frac{f(m-1,n)}2\right\rfloor+\left\lfloor\frac{f(m-1,n-1)}2\right\rfloor+\left\lfloor\frac{f(m-1,n+1)}2\right\rfloor.$$
Show that some positive integer $M$ satisfies $f(M,n)=1$ for $|n|\le(5^{2003}-1)/2$ and $f(M,n)=0$ otherwise.
Editorial clarification: the printed condition $n\in\mathbb N$ is preserved above. It omits negative initial indices and, if zero belongs to $\mathbb N$, conflicts with $f(0,0)$. The intended well-posed initial condition used for the proof is $f(0,n)=0$ for every integer $n\ne0$. The bracket notation in the source is read as floor.