# Periodicity of oscillators in Langton's Ant and powers of $2$

This question based on previous one by me. As Christopher Purcell noticed in his comment, there exist conjecture (which has a lot of counterexamples) that if you take a pair of ants $$(n,n+1)$$ apart (of the same colour, facing the same direction) then you will get an oscillating pattern.

Obviously, that if we place ants too far and areas where they oscillate do not intersects, each ant build a highway. If we work with ants which look in vertical direction and denote their coordinates as $$(x_1,y_1)$$ and $$(x_2,y_2)$$, $$a=|x_1-x_2|$$, $$b=|y_1-y_2|$$, $$n,m$$ - nonnegative integer so ants:

• meet in one cell (and destroy each other) for some $$a=2n, b=2m$$

• create an oscillator for some $$a=2n, b=2m+1$$ and $$a=2n+1, b=2m$$

• build highways for any $$a=2n+1, b=2m+1$$ (and for some $$a=2n, b=2m$$, $$a=2n, b=2m+1$$ and $$a=2n+1, b=2m$$)

If we denote $$p_1$$ as periodicity of any oscillator with $$a=2n+1, b=2m$$ (and $$p_2$$ for any with $$a=2n, b=2m+1$$), $$k$$ - integer, so:

• $$\gcd(p_1,2^k)=4$$ for any $$k>1$$

• $$\gcd(p_2,2^k)\geqslant8$$ for any $$k>2$$

Is there any explanation of this phenomenon?