Summary of Euler lengths for small powers

For comments on the following table, see below.

x \ m=	3	4	5	6	7	8	9	10	11	12
4	5	16	19	15	66	121	146	199	277	911
5	7	5	18	29	39	80	32	188	137	365
6	3	6	14	22	28	55	102	144	145	456
7	4	6	12	21	26	40	67	122	51	196
8	5	16	9	15	21	44	63	89	88	157
9	3	6	11	19	20	34	46	56	70	118
10	5	5	9	15	18	40	32	56	85	66
11	5	7	13	17	22	40	37	44	75	79
12	3	6	6	15	22	34	42	34	61	66
13	4	6	12	16	22	38	33	46	52	78
14	4	6	8	14	20	36	34	39	38	66
15	5	5	10	15	21	36	29	23	46	66
16	5	16	9	13	14	34	29	34	40	54
17	5	6	8	12	17	17	29	34	46	53
18	3	6	10	11	14	34	27	34	39	44
19	3	6	10	11	14	29	26	33	38	42
20	3	5	9	13	17	34	24	34	42	46
21	4	6	10	14	16	27	16	23	39	42
22	5	7	10	12	17	33	25	33	35	44
23	4	6	7	13	14	29	19	32	36	40
24	3	6	6	13	16	33	19	34	35
25	3	5	9	12	13	32	20	26	31
26	4	6	7	12	15		15
27	3	6	9	12	13		18
28	4	6	8	12	12		21
29	3	6	9	12	15		21
30	3	5	6	11	13		17
31	4	5	9	11	15		19
32	4	16	6	11	13		18
33	4	6	8	11	14		19
34	4	6	7	11	13		19
35	4	5	7	13	14		18
36	3	6	6	11	13		17
37	4	6	8	11	13		17
38	3	6	7	11	14		18
39	4	6	8	10	12		19
40	3	5	7	11	12		18
41	3	6	7	11
42	3	6	7	11
43	4	6	9	11
44	3	7	8	11
45	3	5	9	10

The above table highlights a variety of points.

The values for m = 3 are consistent with the conjecture that G(3) £ 4 (see Chapter 7). This is slightly misleading as the above table gives the shortest representation of x^m in terms of lesser mth powers, whereas with g(m) and G(m), x^m is itself the shortest.

For m = 4, the values indicate that when x is a power of 2, at least 16 powers are required, consistent with Davenport’s result that G(4) = 16, and otherwise, that at most 6 are needed in the long term. The trends for higher powers are similar in that they all quickly settle around a plateau of values. One wonders how close these values come to suggesting G(m) ?

The table also suggests that m = 9 in an inherently "better" power for obtaining Euler representations than m = 8, with solutions being found quicker and of shorter length on average for a particular base. However, as seen in Chapter 1, the m = 8 case allows additional restrictive conditions to be applied when searching for the shortest possible.

On a more speculative note, there are vague patterns and instances that suggest the following:

For each m,

are there "favoured" values for Euler lengths, that recur regularly ?

is there any significance in common Euler lengths for consecutive bases ?

is there any significance in exceptional values such as that for 17⁸ ?