Manuscript Oai279-A@1 · Oai279-A.tex · section · a1abc73f

A finite arithmetic table

Stage 6: the explicit circuit for the frozen input › A finite arithmetic table · lines 804–844

For $2 \leq m < 16$ , let $F (m)$ be the complete sorted prime list. The following table supplies $F (m)$ , the set $Z_{m}$ of input bit positions where $m$ has a zero, and the set $T_{m}$ of output wires to be toggled.

$m$	Binary $x_{3} x_{2} x_{1} x_{0}$	$F (m)$	$Z_{m}$	$T_{m}$	Gates
2	0010	$(2)$	${0, 2, 3}$	${5}$	13
3	0011	$(3)$	${2, 3}$	${4, 5}$	12
4	0100	$(2, 2)$	${0, 1, 3}$	${5, 133}$	14
5	0101	$(5)$	${1, 3}$	${4, 6}$	12
6	0110	$(2, 3)$	${0, 3}$	${5, 132, 133}$	13
7	0111	$(7)$	${3}$	${4, 5, 6}$	11
8	1000	$(2, 2, 2)$	${0, 1, 2}$	${5, 133, 261}$	15
9	1001	$(3, 3)$	${1, 2}$	${4, 5, 132, 133}$	14
10	1010	$(2, 5)$	${0, 2}$	${5, 132, 134}$	13
11	1011	$(11)$	${2}$	${4, 5, 7}$	11
12	1100	$(2, 2, 3)$	${0, 1}$	${5, 133, 260, 261}$	14
13	1101	$(13)$	${1}$	${4, 6, 7}$	11
14	1110	$(2, 7)$	${0}$	${5, 132, 133, 134}$	12
15	1111	$(3, 5)$	$\emptyset$	${4, 5, 132, 134}$	10

The output addresses follow (eq:output-address). For example, the first three entries begin at wires $4, 132, 260$ . Their low-bit encodings of $(2, 2, 3)$ have ones at $5, 133, 260, 261$ .

Nearby in this section

lemma: Lemma: Arithmetic validity of the table