# Is Somos-8 $\mod 2$ periodic?

It is known that the Somos-$k$ sequences for $k\ge 8$ do not give integers. But the first terms of Somos-8 sequence $s_n=a_n/b_n$ $$1, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 25, 61, 187, 775, 5827, 14815,\frac{420514}{7}, \frac{28670773}{91}$$ defined by $s_1=s_2=s_3=s_4=s_5=s_6=s_7=s_8=1$, $$s_{n+8}s_n=s_{n+7}s_{n+1}+s_{n+6}s_{n+2}+s_{n+5}s_{n+3}+s_{n+4}^2\qquad(n\ge1)$$ have only odd denominators $b_n$. Morever $s_n$ has even numerator $a_n$ only for $n=9k$. It was checked for $n\le 67$. First terms of $s_n\mod 8:=a_n\cdot b_n^{-1}\mod 8$ are $$\begin{array}{l} 1, 1, 1, 1, 1, 1, 1, 1, {\bf 4},\\ 7, 5, 1, 5, 3, 7, 3, 7, {\bf 6},\\ 7, 7, 5, 7, 5, 3, 1, 1, {\bf 6},\\ 5, 1, 5, 5, 3, 3, 7, 1, {\bf 4},\\ 7, 7, 3, 3, 7, 7, 3, 1, {\bf 2},\\ 1, 3, 1, 3, 7, 5, 3, 5, {\bf 2},\\ 5, 7, 3, 7, 7, 3, 7, 3, {\bf 0},\\ 1, 3, 7, 3,\ldots\end{array}$$

Is it possible to prove that the sequence $s_n\mod 2$ is periodic?

EDT. It was found by მამუკა ჯიბლაძე that numerator of $s_{71}$ is even while that of $s_{72}$ is odd. The last line of the table above is $$1, 3, 7, 3, 1, 5, 1, {\bf 2}, 5$$ So the conjecture about numerators of $s_n$ is wrong. But the qustion about periodicity is still valid.

• There might be something in Robinson, Periodicity of Somos sequences, Proc Amer Math Soc 116 (1992) no. 3, 613-619, MR1140672 (93a:11012). The review by Peter Kiss says, in part, "For some sequences related to the Somos($k$) sequences, the author shows that they are periodic modulo $m$ for every $m$, although the periodicity problem remains open for the original sequences. Some observations are also made concerning the prime divisors of the terms and the rate of growth of certain sequences. Commented Jun 9, 2017 at 6:54
• Believe it or not but actually the numerator of $s_{71}$ is even while that of $s_{72}$ is odd (unless there is a bug in Mathematica 11, that is). Commented Jun 9, 2017 at 7:51
• Doing some further calculations with 2-adics; although I am not sure I am doing it right but seems like $s_{95}$ has even denominator... Commented Jun 9, 2017 at 8:51
• Still have to re-check it but I switched to Pari and computed in 2-adics (with $2^{100000}$ precision), seemingly $s_{103}$ has even denominator. Up to that, results with Mathematica and Pari are identical, and also coincide with results for exact rationals (these I am currently running, they are at 82 so far). Commented Jun 9, 2017 at 11:17
• Alas, run out of memory on 85 with exact rationals. There might be other ways to find out about parity of the denominator in $s_{103}$ though... Commented Jun 10, 2017 at 9:31

I confirm the observation of @მამუკაჯიბლაძე that $$\nu_2(s_{103})=-1$$. That is, $$s_n\bmod 2$$ is not well-defined at first place, which invalidates the question.
Moreover, for $$n\geq 133$$, $$\nu_2(s_n)$$ seems to form a strictly decreasing function, i.e., $$s_n$$ accumulates larger and larger powers of $$2$$ in the denominators.