Prime factor exponent notation

Last updated February 20, 2022

In his 1557 work The Whetstone of Witte , British mathematician Robert Recorde proposed an exponent notation by prime factorisation, which remained in use up until the eighteenth century and acquired the name Arabic exponent notation. The principle of Arabic exponents was quite similar to Egyptian fractions; large exponents were broken down into smaller prime numbers. Squares and cubes were so called; prime numbers from five onwards were called sursolids.

Although the terms used for defining exponents differed between authors and times, the general system was the primary exponent notation until René Descartes devised the Cartesian exponent notation, which is still used today.

This is a list of Recorde's terms.

Cartesian index	Arabic index	Recordian symbol	Explanation
1	Simple
2	Square (compound form is zenzic)	z
3	Cubic	&
4	Zenzizenzic (biquadratic)	zz	square of squares
5	First sursolid	sz	first prime exponent greater than three
6	Zenzicubic	z&	square of cubes
7	Second sursolid	Bsz	second prime exponent greater than three
8	Zenzizenzizenzic (quadratoquadratoquadratum)	zzz	square of squared squares
9	Cubicubic	&&	cube of cubes
10	Square of first sursolid	zsz	square of five
11	Third sursolid	csz	third prime number greater than 3
12	Zenzizenzicubic	zz&	square of square of cubes
13	Fourth sursolid	dsz
14	Square of second sursolid	zbsz	square of seven
15	Cube of first sursolid	&sz	cube of five
16	Zenzizenzizenzizenzic	zzzz	"square of squares, squaredly squared"
17	Fifth sursolid	esz
18	Zenzicubicubic	z&&
19	Sixth sursolid	fsz
20	Zenzizenzic of first sursolid	zzsz
21	Cube of second sursolid	&bsz
22	Square of third sursolid	zcsz

By comparison, here is a table of prime factors:

1 − 20
1	unit
2	2
3	3
4	2²
5	5
6	2·3
7	7
8	2³
9	3²
10	2·5
11	11
12	2²·3
13	13
14	2·7
15	3·5
16	2⁴
17	17
18	2·3²
19	19
20	2²·5

21 − 40
21	3·7
22	2·11
23	23
24	2³·3
25	5²
26	2·13
27	3³
28	2²·7
29	29
30	2·3·5
31	31
32	2⁵
33	3·11
34	2·17
35	5·7
36	2²·3²
37	37
38	2·19
39	3·13
40	2³·5

41 − 60
41	41
42	2·3·7
43	43
44	2²·11
45	3²·5
46	2·23
47	47
48	2⁴·3
49	7²
50	2·5²
51	3·17
52	2²·13
53	53
54	2·3³
55	5·11
56	2³·7
57	3·19
58	2·29
59	59
60	2²·3·5

61 − 80
61	61
62	2·31
63	3²·7
64	2⁶
65	5·13
66	2·3·11
67	67
68	2²·17
69	3·23
70	2·5·7
71	71
72	2³·3²
73	73
74	2·37
75	3·5²
76	2²·19
77	7·11
78	2·3·13
79	79
80	2⁴·5

81 − 100
81	3⁴
82	2·41
83	83
84	2²·3·7
85	5·17
86	2·43
87	3·29
88	2³·11
89	89
90	2·3²·5
91	7·13
92	2²·23
93	3·31
94	2·47
95	5·19
96	2⁵·3
97	97
98	2·7²
99	3²·11
100	2²·5²

External links (references)

Mathematical dictionary, Chas Hutton, pg 224

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Prime factor exponent notation

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