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 Feb 8 awarded Guru Nov 25 comment A definition of Conway base-13 function @MarcelT. Multiplication by 3 (of decimal expansions) is computable, in that for any approximation to the input $x$ which is accurate to precision $\epsilon$ (e.g. if you truncate the decimal expansion), I can produce a value which is within $3\epsilon$ of $x \times 3$. Therefore, any output precision is achievable. However, you cannot do this with $f$: because unless you provide me a promise on the distibution of $p$ and $m$ digits in the input $x$, I cannot know that all of the $p$s and $m$s in the input have been given. So, there is no way for me even to approximate $f(x)$. Jul 29 awarded Yearling Jul 2 awarded Enlightened Jul 2 awarded Nice Answer Apr 11 comment “Consecutive” square residues in odd-order finite fields @JyrkiLahtonen: Thanks! I found a yet simpler, turn-the-crank sort of proof once it occurred to me that the answer might possibly be a straightforward extension of the standard result (and that the standard result might not only be well-known but might also itself be proven simply). Apr 11 answered “Consecutive” square residues in odd-order finite fields Apr 10 asked “Consecutive” square residues in odd-order finite fields Apr 7 comment Orthogonal matrices over GF(2^t) with the first column fixed @JyrkiLahtonen: So, my original motivation is in fact to study orthogonal matrices over GR(q,q^t) for q a power of 2... simplifying to GF(2^t) might have opened up new solutions which do not easily Hensel lift. Do you think it might be best for me to roll back my question? Apr 7 comment Orthogonal matrices over GF(2^t) with the first column fixed @JyrkiLahtonen: Thank you. I think that one might need at least 4x4 blocks because of the case of vectors $\mathbf v$ containing only 1s and 0s, to produce 1s in pairs. (I'm using a similar technique to deal with "unitary" matrices over Galois Rings which quadratically extend other Galois Rings.) This same trick may indeed work for GF(2^t), but sadly it is not clear that it would Hensel lift to orthogonal matrices over GR(q,q^t) for q a power of 2, unless it so happened that all of the other columns were unit vectors mod 4 as well as mod 2. Apr 7 revised Orthogonal matrices over GF(2^t) with the first column fixed Minor revisions to improve presentation, as well as to remove redundant information left over from first version Apr 6 revised Orthogonal matrices over GF(2^t) with the first column fixed Focused on a special case likely to give rise to an answer to the more general question Apr 6 asked Orthogonal matrices over GF(2^t) with the first column fixed Mar 13 revised Cayley-Dickson construction: a general rule for multiplying imaginary units? Revised description (the formula is no longer "closed form" in presentation), removed redundant description of $\oplus$ Mar 13 revised Cayley-Dickson construction: a general rule for multiplying imaginary units? Corrected the formula for $N_{x,y}$ Mar 11 revised Cayley-Dickson construction: a general rule for multiplying imaginary units? Updated in light of my answer Mar 11 answered Cayley-Dickson construction: a general rule for multiplying imaginary units? Mar 6 asked Cayley-Dickson construction: a general rule for multiplying imaginary units? Mar 5 revised A puzzle on orthogonal matrices modulo $p$ Removed reference to higher dimensional examples after discovery of some complications Mar 5 awarded Excavator