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Suppose $x_1,x_2,...,x_b$ is a Complete Residue System (CRS) mod $b$.

is there a CRS say $y_1,y_2,...,y_b$ s.t. $\vert y_i \vert \leq \frac{b}{2}$ $\forall i = 1,2,...,b$

I want to say you CAN find one, namely, does the following work for mod $4$:

take $x_1,x_2,x_3,x_4$ be $\{4,9,14,19\}$

then $4 \in [0]_4$, $9 \in [1]_4$, $14 \in [2]_4$, $19 \in [3]_4$

I claim $y_1,y_2,...,y_b$ is $\{0,1,2,-1\}$

which is still a CRS, has 4 elements AND

$0 \in [0]_4$, $1 \in [1]_4$, $2 \in [2]_4$, $-1 \in [3]_4$

Making $y_1,y_2,...,y_b$ = $\{0,1,2,-1\}$ a CRS modulo 4.

Thanks in advance!!!

homosapien
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    Use a least magnitude residue system $,0, \pm1, \pm 2,\ldots \ \ $ – Bill Dubuque Apr 13 '19 at 22:09
  • so what you're saying is to show that such a CRS exists to use a more simple example? instead of 4, just using 1 or 2 works as well – homosapien Apr 14 '19 at 15:43
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    Yes, there is a least magnitude CRS for any modulus (convention: choose $n$ (vs. $-n)$ modulo $m = 2n$, e.g. $,0,\pm1,\pm2, 3\pmod{6}.\ $ Note that this is a sequence of $m$ consecutive integers (balanced around $0$) and any sequence iof $m$ consecutive integers is a CRS mod $m$ (just a shift of the standard CRS). – Bill Dubuque Apr 14 '19 at 15:54
  • ohh wow gotcha, so this new CRS $y_1,...,y_b$ is a "shift" if you will of my original one by a certain magnitude? so there's always a least magnitude for any given modulus . THANKS! – homosapien Apr 14 '19 at 16:01
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    Yes, e.g. see here. – Bill Dubuque Apr 14 '19 at 16:04

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