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I'd like to ask how to solve the quadratic nimber equation $x\otimes x \oplus b \otimes x \oplus c=0$, where $\otimes$ is nim multiplication and $\oplus$ is nim addition.

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From now on, all arithmetic is nimber arithmetic, except $2^i$ will refer to the usual power of two. When $b=0$, the unique solution is $x=\sqrt{c}$, so assume $b\neq 0$.

In order to solve $z^2+bz+d=0$, use the substitution $z=bx$ to reduce this to $$ x^2+x=c. $$ where $c=d/b^2$. That is, letting $f(x)=x^2+x$, you need $f^{-1}(c)$.

Note that $f$ is a linear map whose kernel is $\{0,1\}$, which means $f$ is a bijection from the subspace of even nimbers to all nimbers. I will use $f^{-1}(y)$ to denote the unique even nimber $x$ for which $f(x)=y$. Note $f^{-1}$ is also a linear map.

More specifically, it can be shown that

Prop: For any $n\ge 1$, $f$ is a surjective map from the interval $[2^n,2^{n+1})$ to $[2^{n-1},2^n)$.

I omit the proof.

Writing $c=\sum_i b_i 2^i$ with $b_i\in \{0,1\}$, we have $f^{-1}(c) = \sum_i b_i f^{-1}(2^i)$, so it suffices to compute $f^{-1}(2^i)$ for any $i$. This can be done recursively as follows. By the above proposition, there are (easily computable) numbers $a_{ij}\in \{0,1\}$ for which $$ f(2^{i+1})=2^i + \sum_{j=0}^{i-1} a_{ij}2^j $$ Therefore, $$ f^{-1}(2^i) = 2^{i+1}+\sum_{j=0}^{i-1}a_{ij}f^{-1}(2^i) $$ so $f^{-1}(2^i)$ can be computed from previously computed values $f^{-1}(2^j)$.

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  • $\begingroup$ There should be two solutions $x_1$, $x_2$ to the equation. Since $x_1 \oplus x_2=b$, we may get another if one is known. But you say $f$ is a bijection? $\endgroup$ – Hang Wu Jan 30 at 2:25
  • $\begingroup$ The two "oblong roots" of any nimber sum to $1$, so one is even and one is odd. Can we define a "principal" oblong root as the even one and thus repair the bijectivity? $\endgroup$ – Oscar Lanzi Jan 30 at 2:58
  • $\begingroup$ @HangWu $f$ is bijective when it’s domain is restricted to the even nimbers. I defined $f^{-1}$ to be the inverse for that restricted function. $\endgroup$ – Mike Earnest Jan 30 at 3:06
  • $\begingroup$ I see. Thanks a lot. $\endgroup$ – Hang Wu Jan 30 at 3:25

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